Right Triangle Trigonometry Today in Sports, Surveying, Navigation,* and Engineering

CHAPTER Right Triangle [ 1 [ Trigonometry

To the ancient Greeks, trigonometry was the study of right triangles. Trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) can be defined as right triangle ratios (ratios of the lengths of sides of a right triangle). Thousands of years later, we still find applications of right triangle trigonometry today in sports, surveying, navigation,* and engineering.

*Section 1.5, Example 7 and Exercises 66–68 and 73–74. VitalyEdush/Getty Images, Inc.

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LEARNING OBJECTIVES

■■Understand degree ■■Define the six trigono- ■■Evaluate trigonometric measure. metric functions as ratios functions exactly and with ■■Learn the conditions that of lengths of the sides of calculators. make two triangles similar. right triangles. ■■Solve right triangles.

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We will review angles, degree measure, and special right triangles. We will discuss the properties of similar triangles. We will use the concept of similar right triangles to define the six trigonometric functions as ratios of the lengths of the sides of right triangles (right triangle trigonometry).

RIGHT TRIANGLE TRIGONOMETRY 1.1 1.2 1.3 1.4 1.5 ANGLES, SIMILAR DEFINITION 1 OF EVALUATING SOLVING RIGHT DEGREES, TRIANGLES TRIGONOMETRIC TRIGONOMETRIC TRIANGLES AND TRIANGLES FUNCTIONS: FUNCTIONS: RIGHT TRIANGLE EXACTLY RATIOS AND WITH CALCULATORS

• Angles and • Finding Angle • Trigonometric • Evaluating • Accuracy and Degree Measures Using Functions: Trigonomet- Significant Measure Geometry Right Triangle ric Functions Digits • Triangles • Classification of Ratios Exactly for • Solving a Right • Special Right Triangles • Cofunctions Special Angle Triangle Given Triangles Measures: 308, an Acute Angle 458, and 608 Measure and • Using Calcula- a Side Length tors to Evaluate • Solving a Right (Approximate) Triangle Given Trigonometric the Lengths of Function Values Two Sides • Representing Partial Degrees: DD or DMS

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1.1 ANGLES, DEGREES, AND TRIANGLES SKILLS OBJECTIVES CONCEPTUAL OBJECTIVES ■■ Find the complement and supplement of an angle. ■■ Understand that degrees are a measure of an angle. ■■ Use the Pythagorean theorem to solve a right triangle. ■■ Understand that the Pythagorean theorem applies only to ■■ Solve 308-608-908 and 458-458-908 triangles. right triangles. ■■ Understand that to solve a triangle means to find all of the angle measures and side lengths.

1.1.1 Angles and Degree Measure

1.1.1 SKILL The study of trigonometry relies heavily on the concept of angles. Before we define Find the complement and angles, let us review some basic terminology. A line is the straight path connecting supplement of an angle. two points (A and B) and extending beyond the points in both directions. The portion of the line between the two points (including the points) is called a line segment. A ray is the portion of the line that starts at one point (A) and extends to infinity 1.1.1 CONCEPTUAL (beyond B). Point A is called the endpoint of the ray. Understand that degrees are a In geometry, an angle is formed when two rays share the same endpoint. The measure of an angle. common endpoint is called the vertex. In trigonometry, we say that an angle is formed when a ray is rotated around its endpoint. The ray in its original position is called A B the initial ray or the initial side of an angle. In the Cartesian plane (the rectangular Line AB coordinate plane), we usually assume the initial side of an angle is the positive x-axis and the vertex is located at the origin. The ray after it is rotated is called the terminal A B ray or the terminal side of an angle. Rotation in a counterclockwise direction Segment AB corresponds to a positive angle, whereas rotation in a clockwise direction corresponds A B to a negative angle. Ray AB Initial side Terminal Negative angle side

Angle Terminal Positive angle Vertex side Initial side

STUDY TIP Lengths, or distances, can be measured in different units: feet, miles, and meters are Positive angle: Counterclockwise three common units. To compare angles of different sizes, we need a standard unit of Negative angle: Clockwise measure. One way to measure the size of an angle is with degree measure. We discuss degrees now, and in Chapter 3, we discuss another angle measure called radians.

CONCEPT CHECK [ ] DEFINITION Degree Measure of Angles TRUE OR FALSE The measure of An angle formed by one complete counterclockwise rotation has measure 360 an acute angle is greater than the measure of an obtuse angle. degrees, denoted 360°. ▼ ANSWER False One complete revolution = 360º

1 Therefore, a counterclockwise 360 of a rotation has measure 1 degree.

WORDS MATH 360° represents 1 complete rotation. 1 complete rotation 5 1⋅360° 5 360° 1 1 1 180° represents 2 rotation. 2 complete rotation 5 2 ⋅360° 5 180° 1 1 1 90° represents 4 rotation. 4 complete rotation 5 4 ⋅360° 5 90°

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The Greek letter u (theta) is the most common name for an angle in mathematics. Other common names for angles are a (alpha), b (beta), and g (gamma). STUDY TIP Greek letters are often used to denote angles in trigonometry. WORDS MATH An angle measuring exactly 90° is called a right angle. A right angle is = 90º 1 often represented by the adjacent θ Right angle: 4 rotation sides of a rectangle, indicating that the two rays are perpendicular.

An angle measuring exactly 180° is θ = 180º 1 called a straight angle. Straight angle: 2 rotation

An angle measuring greater than 08, but less than 90°, is called an acute angle. Acute angle 0º < θ < 90º θ

An angle measuring greater than 90°, but less than 180°, is called an obtuse angle. θ Obtuse angle 90º < θ < 180º

If the sum of the measures of two positive angles is 90°, the angles are Complementary angles β called complementary. We say that a is α + β = 90º the complement of b (and vice versa). α

If the sum of the measures of two positive angles is 180°, the angles are α Supplementary angles called supplementary. We say that a is β α + β = 180º the supplement of b (and vice versa).

EXAMPLE 1 Finding Measures of Complementary and Supplementary Angles Find the measure of each angle: a. Find the complement of 50°. b. Find the supplement of 110°. c. Represent the complement of a in terms of a. d. Find two supplementary angles such that the first angle is twice as large as the second angle. Solution: a. The sum of complementary angles is 90°. u 1 50° 5 90° Solve for u. u 5 40°

b. The sum of supplementary angles is 180°. u 1 110° 5 180° Solve for u. u 5 70°

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c. Let b be the complement of a. The sum of complementary angles is 90°. a 1 b 5 90° Solve for b. b 5 90° 2 a

d. The sum of supplementary angles is 180°. a 1 b 5 180° Let b 5 2a. a 1 2a 5 180° Solve for a. 3a 5 180° a 5 60° Substitute a 5 60° into b 5 2a. b 5 120°

The angles have measures 60° and 120° .

▼ ▼ ANSWER YOUR TURN Find two supplementary angles such that the first angle is three The angles have times as large as the second angle. measures 45° and 135°. 1.1.2 Triangles

1.1.2 SKILL Trigonometry originated as the study of triangles, with emphasis on calculations Use the Pythagorean theorem to involving the lengths of sides and the measures of angles. Triangles are three-sided solve a right triangle. closed-plane figures. An important property of triangles is that the sum of the measures of the three angles of any triangle is 180°.

1.1.2 CONCEPTUAL Understand that the Pythagorean theorem applies only to right ANGLE SUM OF A TRIANGLE triangles. The sum of the measures of the angles of any triangle is 180°.

α + β + γ = 180º

α β

EXAMPLE 2 Finding an Angle of a Triangle If two angles of a triangle have measures 32° and 68°, what is the measure of the third angle?

32º 68º

Solution: The sum of the measures of all three angles is 180°. 32° 1 68° 1 a 5 180° Solve for a. a 5 80°

▼ ▼ ANSWER YOUR TURN If two angles of a triangle have measures 16° and 96°, what is the 68° measure of the third angle?

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In geometry, some triangles are classified as equilateral, isosceles, and right. An equilateral triangle has three equal sides and therefore has three equal angles 16082. An isosceles triangle has two equal sides (legs) and therefore has two equal angles opposite those legs. The most important triangle that we will discuss in this course is a right triangle. A right triangle is a triangle in which one of the angles is a right angle 190°2. Since one angle is 90°, the other two angles must be complementary 1sum to 90°2 so that the sum of all three angles is 180°. The longest side of a right triangle, called the hypotenuse, is opposite the right angle. The other two sides are called the legs of the right triangle.

Hypotenuse

Right triangle: Leg STUDY TIP In this book when we say “equal Leg angles,” this implies “equal angle measures.” Similarly, when we say an angle is x8, this implies The Pythagorean theorem relates the sides of a right triangle. It is important to that the angle’s measure is x8. note that length (a synonym of distance) is always positive.

PYTHAGOREAN THEOREM In any right triangle, the square of the length of the longest side (hypotenuse) is [ CONCEPT CHECK] equal to the sum of the squares of the lengths of the other two sides (legs). TRUE OR FALSE The hypotenuse is always longer than each of the legs of a right triangle. ▼ ANSWER True c a2 1 b2 5 c2 a

It is important to note that the Pythagorean theorem applies only to right triangles. In addition, it does not matter which leg is called a or b as long as the square of the longest side is equal to the sum of the squares of the smaller sides.

EXAMPLE 3 Using the Pythagorean Theorem to Find the Side of a Right Triangle Suppose you have a 10-foot ladder and want to reach a height of 8 feet to clean out the gutters on your house. How far from the base of the house should the base of the ladder be? Solution: 10 ft 8 ft

Label the unknown side as x. 10 8 ?

Apply the Pythagorean theorem. x2 1 82 5 102 Simplify. x2 1 64 5 100

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Solve for x. x2 5 36 x 5 66 Length must be positive. x 5 6

The ladder should be 6 feet from the base of the house along the ground.

▼ ▼ ANSWER YOUR TURN A steep ramp is being built for skateboarders. The height, 15 ft horizontal ground distance, and ramp length form a right triangle. If the height is 9 feet and the horizontal ground distance is 12 feet, what is the length of the ramp?

When solving a right triangle exactly, simplification of radicals is often necessary. For example, if a side length of a triangle resulted in !17, the radical cannot be simplified any further. However, if a side length of a triangle resulted in !20, the radical would be simplified:

"20 5 "4⋅5 5 "4⋅"5 5 "22 ⋅"5 5 2"5

EXAMPLE 4 Using the Pythagorean Theorem with Radicals Use the Pythagorean theorem to solve for the unknown side length in the given right triangle. Express your answer exactly in terms of simplified radicals. Solution: Apply the Pythagorean theorem. 32 1 x2 5 72

2 Simplify known squares. 9 1 x 5 49 7 x Solve for x. x2 5 40 x 5 6"40 3 The side length x is a distance that is positive. x 5 "40 Simplify the radical. x 5 "4⋅10 5 "4⋅"10 5 2"10 2f ▼ ▼ ANSWER YOUR TURN Use the Pythagorean theorem to solve for the 4"3 unknown side length in the given right triangle. Express your answer exactly in terms of 8 simplified radicals. x

1.1.3 Special Right Triangles

1.1.3 SKILL Right triangles whose sides are in the ratios of 3-4-5, 5-12-13, and 8-15-17 are examples Solve 308-608-908 and of right triangles that are special because their side lengths are equal to whole numbers 458-458-908 triangles. that satisfy the Pythagorean theorem. A Pythagorean triple consists of three positive integers that satisfy the Pythagorean theorem. 1.1.3 CONCEPTUAL 32 1 42 5 52 52 1 122 5 132 82 1 152 5 172 Understand that to solve a triangle There are two other special right triangles that warrant attention: a 308-608-908 triangle means to find all of the angle and a 458-45°-908 triangle. Although in trigonometry we focus more on the angles than measures and side lengths.

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on the side lengths, we are interested in special relationships between the lengths of the sides of these right triangles. We will start with a 458-458-908 triangle.

WORDS MATH A 458-458-908 triangle is an isosceles (two legs are equal) right triangle. 45º x

45º x

Apply the Pythagorean theorem. x2 1 x2 5 hypotenuse2 Simplify the left side of the equation. 2x2 5 hypotenuse2

2 Solve for the hypotenuse. hypotenuse 5 6"2x 5 6!2 0 x0 The x and the hypotenuse are both lengths and, therefore, must be positive. hypotenuse 5 !2x This shows that the hypotenuse of a 458-458-908 is !2 times the length of 45º either leg. √2x x

45º x

If we let x 5 1, then the triangle will have legs with length equal to 1 and the hypotenuse will have length !2. Notice that these lengths satisfy the Pythagorean 2 2 2 theorem: 1 1 1 5 A !2 B , or 2 5 2. Later when we discuss the unit circle approach, we !2 !2 will let the hypotenuse have length 1. The legs will then have lengths and : 2 2 !2 2 !2 2 1 1 a b 1 a b 5 1 5 12 2 2 2 2

EXAMPLE 5 Solving a 458-458-908 Triangle A house has a roof with a 45° pitch (the angle the roof makes with the house). If the house is 60 feet wide, xx what are the lengths of the sides of the roof that form the attic? Round to the nearest foot. 45º 45º 60 ft

Solution: Draw the 45°-45°-908 triangle. Let x represent the length of the unknown legs. xx

45º 45º 60 ft

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Recall that the hypotenuse of a 45°-45°-908 STUDY TIP triangle is !2 times the length of either leg. “To solve a triangle” means to xx find all of the angle measures and side lengths. 45º 45º √2x

[ CONCEPT CHECK] Let the hypotenuse equal 60 feet. 60 5 2x TRUE OR FALSE In a right triangle, ! 42 ft 42 ft if you know one of the acute angles 60 and one side length, then it is Solve for x. x 5 < 42.42641 possible to solve the triangle. !2 45º 45º 60 ft ▼ Round to the nearest foot. x < 42 ft ANSWER True

▼ ▼ ANSWER YOUR TURN A house has a roof with a 45° pitch. If the sides of the roof are 60!2 < 85 ft 60 feet, how wide is the house? Round to the nearest foot.

Let us now determine the relationship of the sides of a 308-608-908 triangle. We start with an equilateral triangle (equal sides and equal 60° angles).

WORDS MATH Draw an equilateral triangle with sides 2x. 60º

2x 2x

60º 60º 2x

Draw a line segment from one vertex that is perpendicular to the opposite side; this line segment represents the height of 30º 30º the triangle h and bisects the base. There are 2x 2x now two identical 308-608-908 triangles. h

60º 60º xx

Notice that in each triangle the hypotenuse is twice the length of the shortest leg, which is opposite the 30° angle. 30º

h 2x

60º x

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To find the length h, use the Pythagorean theorem. h2 1 x2 5 12x22 Solve for h. h2 1 x2 5 4x2 h2 5 3x2

h 5 6"3x2 5 6!3 0 x 0 Both h and x are lengths and must be positive. h 5 !3x

We see that the hypotenuse of a 308-608-908 triangle is two times the length of the leg opposite the 30° angle, 30º the short side. 2x √3x The leg opposite the 60° angle, the long leg, is !3 times the length of the leg opposite the 30° angle, the short leg. 60º x

If we let x 5 1, then the triangle will have legs with lengths 1 and !3 and hypotenuse of length 2. These lengths satisfy the Pythagorean theorem: 12 1 1 !322 5 22, or 4 5 4. Later when we discuss the unit circle approach, we will let the hypotenuse have length 1. 1 !3 The legs will then have lengths and . 2 2 1 2 !3 2 1 3 a b 1 a b 5 1 5 12 2 2 4 4

EXAMPLE 6 Solving a 308-608-908 Triangle Before a hurricane strikes, it is wise to stake down trees for additional support during the storm. If the branches allow for the rope to be tied 15 feet up the tree and a desired angle between the rope and the ground is 60°, how much total rope is needed? How far from the base of the tree should each of the two stakes be hammered? 30º 15 ftf

60º 60º

Solution: Recall the relationship between the sides of a 308-608-908 triangle. 30º Label each side. 2x √3x = 15 ft In this case, the leg opposite the 60° angle is 15 feet. 60º x

Solve for x. !3x 5 15 ft 15 x 5 < 8.7 ft !3

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15 Find the length of the hypotenuse. hypotenuse 5 2x 5 2 a b < 17 ft !3

The ropes should be staked approximately 8.7 feet from the base of the tree, and approximately 21172 5 34 total feet of rope will be needed. 30 17 ft 17 ft

60º 60º 8.7 ft

▼ ▼ ANSWER YOUR TURN Rework Example 6 with a height (where the ropes are tied) of Each rope should be staked 20 feet. How far from the base of the tree should each of the ropes approximately 12 feet from the base of be staked, and how much total rope will be needed? the tree. Approximately 46 total feet of rope will be needed.

[SECTION 1.1] SUMMARY

In this section, you have practiced working with angles and triangles. One unit of measure of angles is degrees. An angle measuring exactly 908 is called a right angle. The sum of the three 30º angles of any triangle is always 1808. Triangles that contain a right 2 angle are called right triangles. With the Pythagorean theorem, 45º √3 √2 you can solve for one side of a right triangle given the other two 1 sides. The right triangles 308-608-908 and 458-458-908 are special 45º 60º because of the simple numerical relations of their sides. 1 1

[SECTION 1.1] EXERCISES

• SKILLS

In Exercises 1–8, specify the measure of the angle in degrees using the correct algebraic sign 11 or 22. 1 1 1 2 1. 2 rotation counterclockwise 2. 4 rotation counterclockwise 3. 3 rotation clockwise 4. 3 rotation clockwise 5 7 4 5 5. 6 rotation counterclockwise 6. 12 rotation counterclockwise 7. 5 rotation clockwise 8. 9 rotation clockwise

In Exercises 9–14, find (a) the complement and (b) the supplement of each of the given angles. 9. 18° 10. 39° 11. 42° 12. 57° 13. 89° 14. 75°

In Exercises 15–18, find the measure of each angle. 15. 16.

(3x)º (15x)º

(6x)º (4x)º

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17. Supplementary angles with measures 8x degrees and 4x degrees 18. Complementary angles with measures 3x 1 15 degrees and 10x 1 10 degrees

In Exercises 19–24, refer to the triangle in the drawing. 19. If a 5 117° and b 5 33°, findg . 20. If a 5 110° and b 5 45°, find g. γ 21. If g 5 b and a 5 4b, find all three angles.

22. If g 5 b and a 5 3b, find all three angles. α 23. If b 5 53.3° and g 5 23.6°, finda . β α + β + γ = 180º 24. If a 5 105.6° and g 5 13.2°, find b.

In Exercises 25–34, refer to the right triangle in the drawing. Express lengths exactly. 25. If a 5 4 and b 5 3, find c. 26. If a 5 3 and b 5 3, find c. 27. If a 5 6 and c 5 10, findb . 28. If b 5 7 and c 5 12, find a. 29. If a 5 8 and b 5 5, find c. 30. If a 5 6 and b 5 5, find c. c a 31. If a 5 7 and c 5 11, findb . 32. If b 5 5 and c 5 9, find a. 33. If b 5 !7 and c 5 5, finda . 34. If a 5 5 and c 5 10, find b. b In Exercises 35–40, refer to the 458-458-908 triangle in the drawing. Express lengths exactly. 35. If each leg has length 10 inches, how long is the hypotenuse? 36. If each leg has length 8 meters, how long is the hypotenuse? 45º 37. If the hypotenuse has length 2!2 centimeters, how long is each leg? √2x x 38. If the hypotenuse has length !10 feet, how long is each leg? 39. If each leg has length 4!2 inches, how long is the hypotenuse? 45º x 40. If the hypotenuse has length 6 meters, how long is each leg?

In Exercises 41–46, refer to the 308-608-908 triangle in the drawing. Express lengths exactly. 41. If the shortest leg has length 5 meters, what are the lengths of the other leg and the hypotenuse?

42. If the shortest leg has length 9 feet, what are the lengths of the other leg and the hypotenuse? 30º 43. If the longer leg has length 12 yards, what are the lengths of the other leg and the hypotenuse? 2x 44. If the longer leg has length n units, what are the lengths of the other leg and the hypotenuse? √3x 45. If the hypotenuse has length 10 inches, what are the lengths of the two legs? 46. If the hypotenuse has length 8 centimeters, what are the lengths of the two legs? 60º x

• APPLICATIONS 47. Clock. What is the measure of the angle (in degrees) that the minute hand traces in 20 minutes? 48. Clock. What is the measure of the angle (in degrees) that the minute hand traces in 25 minutes? 49. London Eye. The London Eye (similar to a bicycle wheel) makes one rotation in approximately 30 minutes. What is the measure of the angle (in degrees) that a cart (spoke) will rotate in 12 minutes? 50. London Eye. The London Eye (similar to a bicycle wheel) makes one rotation in approximately 30 minutes. What is the measure of the angle (in degrees) that a cart (spoke) will rotate in 5 minutes? 51. Revolving Restaurant. If a revolving restaurant can rotate 270° in 45 minutes, how long does it take for the restaurant to

make a complete revolution? David Ball/Getty Images, Inc.

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52. Revolving Restaurant. If a revolving restaurant can rotate 59. Tree Stake. A tree needs to be staked down before a storm. 72° in 9 minutes, how long does it take for the restaurant to If the ropes can be tied on the tree trunk 10 feet above the make a complete revolution? ground and the staked rope should make a 30° angle with the 53. Field Trial. In a Labrador retriever field trial, a dog is judged ground, how far from the base of the tree should each rope be by the straightness of the line it takes to retrieve a fallen bird. staked? The competitors are required to go through the water, not along 60. Tree Stake. What is the total amount of rope (in feet) that the shore. If the judge wants to calculate how far a dog will extends to the four stakes supporting the tree in Exercise 59? travel along a straight path, she walks the two legs of a right 61. Party Tent. Steve and Peggy want to rent a 40-foot by triangle as shown in the drawing and uses the Pythagorean 20-foot tent for their backyard to host a barbecue. The base theorem. How far would this dog travel (run and swim) if it of the tent is supported 7 feet above the ground by poles, and traveled along the hypotenuse? Round to the nearest foot. then roped stakes are used for support. The ropes make a 60° angle with the ground. How large a footprint in their yard would they need for this tent (and staked ropes)? In other words, what are the dimensions of the rectangle formed by the stakes on the ground? Round to the nearest foot. 80 ft Pond

30 ft

54. Field Trial. How far would the dog in Exercise 53 run and swim if it traveled along the hypotenuse if the judge walks 25 feet along the shore and then 100 feet out to the bird. Round to the nearest foot.

100 ft 62. Party Tent. Ashley’s parents are throwing a graduation party and are renting a 40-foot by 80-foot tent for their backyard. Pond The base of the tent is supported 7 feet above the ground by poles, and then roped stakes are used for support. The ropes make a 45° angle with the ground. How large a footprint (see 25 ft Exercise 61) in their yard will they need for this tent (and 55. Christmas Lights. A couple want to put up Christmas lights staked ropes)? Round to the nearest foot. along the roofline of their house. If the front of the house is 63. Fence Corner. Ben is checking to see if his fence measures 100 feet wide and the roof has a 45° pitch, how many linear 90° at the corner. To do so, he measures out 10 feet along feet of Christmas lights should the couple buy? Round to the the fence and places a stake marking it point A. He then goes nearest foot. back to the corner and measures out 15 feet along the fence in the other direction and places a stake marking it point B as shown in the figure below. Finally, he measures the distance between points A and B and finds it to be 20 feet. Does his corner measure 90°? 45º 45º 100 ft B

20 ft

56. Christmas Lights. Repeat Exercise 55 for a house that is 15 ft 60 feet wide. Round to the nearest foot. 57. Tree Stake. A tree needs to be staked down before a storm. If the ropes can be tied on the tree trunk 17 feet above the ground and the staked rope should make a 60° angle with the 10 ft A ground, how far from the base of the tree should each rope be staked? 64. Fence Corner. For another corner of Ben’s fence (see Exercise 58. Tree Stake. What is the total amount of rope (in feet) that 63), he measures from the corner up one side of the fence extends to the two stakes supporting the tree in Exercise 57? 8 feet and marks it with a stake as point A. He then measures

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diagonally across the yard 17 feet from point A to the fence going in the other direction as shown and marks it as point B. If this corner is 90°, what is the length of the fence from the corner to point B?

A 65. Car Engine. Bill’s car engine is said to run at 1700 RPMs 17 ft (revolutions per minute) at idle. Through how many degrees does his engine turn each second? 8 ft 66. Car Engine. Upon acceleration, Bill’s car engine turns 300,000° in 15 seconds. At what RPM is the car engine turning?

• CATCH THE MISTAKE In Exercises 67 and 68, explain the mistake that is made. 67. In a 308-608-908 triangle, find the length of the side opposite 68. In a 458-458-908 triangle, find the length of the hypotenuse if the 60° angle if the side opposite the 30° angle is 10 inches. each leg has length 5 centimeters.

Solution: Solution: 2 2 2 The length opposite the 60° angle is Use the Pythagorean theorem. 5 1 5 5 hypotenuse twice the length opposite the 30° angle. 21102 5 20 Simplify. 50 5 hypotenuse2 The side opposite the 60° angle has length Solve for the hypotenuse. hypotenuse 5 65!2 20 inches. The hypotenuse has length 65!2 centimeters. This is incorrect. What mistake was made? This is incorrect. What mistake was made?

• CONCEPTUAL In Exercises 69–76, determine whether each statement is true or false. 69. The Pythagorean theorem can be applied to any equilateral 73. The two acute angles in a right triangle must be complemen- triangle. tary angles. 70. The Pythagorean theorem can be applied to all isosceles 74. In a 458-458-908 triangle, the length of each side is the triangles. same. 71. The two angles opposite the legs of a right triangle are 75. Angle measure in degrees can be both positive and negative. complementary. 76. A right triangle cannot contain an obtuse angle. 72. In a 308-608-908 triangle, the length of the side opposite the 60° angle is twice the length of the side opposite the 30° angle.

• CHALLENGE 77. What is the measure (in degrees) of the smaller angle the hour 81. If AC 5 30, AB 5 24, and DC 5 11, find AD. and minute hands form when the time is 12:20? 82. If AB 5 60, AD 5 61, and DC 5 36, find AC. 78. What is the measure (in degrees) of the smaller angle the hour 83. Given a square with side length x, draw the two diagonals. and minute hands form when the time is 9:10? The result is four special triangles. Describe these triangles. In Exercises 79–82, use the figure below: What are the angle measures? A 84. Solve for x in the triangle below.

3x – 2 2x + 2

B D C

79. If AB 5 3, AD 5 5, and AC 5 !58, find DC. x 80. If AB 5 4, AD 5 5, and AC 5 !41, find DC.

• TECHNOLOGY 85. If the shortest leg of a 308-608-908 triangle has length 16.68 feet, 86. If the longer leg of a 308-608-908 triangle has length what are the lengths of the other leg and the hypotenuse? Round 134.75 centimeters, what are the lengths of the other leg and answers to two decimal places. the hypotenuse? Round answers to two decimal places.

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1.2 SIMILAR TRIANGLES SKILLS OBJECTIVES CONCEPTUAL OBJECTIVES ■■ Find angle measures using geometry. ■■ Understand the properties of angles and how to apply ■■ Use similarity to determine the length of a side of those properties. a triangle. ■■ Understand the difference between similar and congruent triangles.

1.2.1 Finding Angle Measures Using Geometry

1.2.1 SKILL We stated in Section 1.1 that the sum of the measures of three angles of any triangle is Find angle measures using 180°. We now review some angle relationships from geometry. geometry. Vertical angles are angles of equal measure that are opposite one another and share the same vertex. In the figure below, angles 1 and 3 are vertical angles. Angles 2 and 4 are also vertical angles. 1.2.1 CONCEPTUAL Understand the properties of angles and how to apply those properties.

1 2 4 3

m||n A transversal is a line that intersects two other coplanar lines. If a transversal 12 intersects two parallel lines m and n, or m i n, the corresponding angles have equal 34 m measure. Angles 3, 4, 5, and 6 are classified as interior angles, whereas angles 1, 2, 7, and 8 are classified as exterior angles. In diagrams, there are two traditional ways to indicate angles having equal measure: 56 78 n ■■ with the same number of arcs, or Transversal ■■ with a single arc and the same number of hash marks. In this text, we will use the same number of arcs.

[ CONCEPT CHECK] PROPERTIES OF ANGLES Assuming you have two parallel NAME PICTURE RULE lines cut by a transversal, which of the following are angles of the Vertical angles Vertical angles have equal same measure: measure. 1 (A) vertical 2 /1 5 /3 and /2 5 /4 (B) alternate interior 4 (C) alternate exterior 3 (D) corresponding (E) all of the above ▼ Alternate interior angles m||n Alternate interior angles have ANSWER (E) all of the above equal measure. 3 4 m /3 5 /6 and /4 5 /5

5 6 n

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NAME PICTURE RULE Alternate exterior angles m||n Alternate exterior angles have equal measure. 1 2 m /1 5 /8 and /2 5 /7

7 8 n

Corresponding angles m||n Corresponding angles have equal measure. 1 2 /1 5 /5 34 m Note that the following are also corresponding angles: 5 6 /2 and /6 78 n /3 and /7 /4 and /8

Interior angles on the same m||n Interior angles on the same side of transversal side of the transversal have measures that sum to 1808 3 4 m (they are supplementary). ∠3 + ∠5 = 180º ∠4 + ∠6 = 180º 5 6 n

EXAMPLE 1 Finding Angle Measures Given that /1 5 110° and m i n, m||n find the measure of angle 7. ∠1 = 110º m

∠7 = ? n

Solution: Corresponding angles have equal measure, m||n /1 5 /5. ∠1 = 110º

∠5 = 110º

/5 and /7 are supplementary angles. m||n

∠5 = 110º ∠7

Measures of supplementary angles sum to 180°. /5 1 /7 5 180° Substitute /5 5 110°. 110° 1 /7 5 180° Solve for /7. /7 5 70°

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Draw two parallel lines (solid) and draw two transverse lines (dashed), which intersect at a point on one of the parallel lines. 2 1 3 The corresponding angles 1 and 6 have equal measure. 4 The vertical angles 2 and 4 have equal measure. 5 6 The corresponding angles 3 and 5 have equal measure.

The sum of the measures of angles 1, 2, and 3 is 180°. Therefore, the sum of the measures of angles 4, 5, and 6 is also 180°. So the sum of the measures of three angles of any triangle is 180°.

1.2.2 Classification of Triangles

1.2.2 SKILL Angles with equal measure are often labeled with the same number of arcs. Similarly, Use similarity to determine the sides of equal length are often labeled with the same number of hash marks. length of a side of a triangle.

1.2.2 CONCEPTUAL NAME PICTURE RULE Understand the difference Equilateral triangle All sides are equal. between similar and congruent triangles.

Isosceles triangle Two sides are equal. [ CONCEPT CHECK] TRUE OR FALSE Two similar triangles are congruent triangles, Scalene triangle No sides are equal. but two congruent triangles need not be similar triangles. ▼ ANSWER False

The word similar in geometry means identical in shape, although not necessarily the same size. It is important to note that two triangles can have the exact same shape (same angles) but be of different sizes.

DEFINITION Similar Triangles Similar triangles are triangles with equal corresponding angle measures (equal angles).

The word congruent means equal in all corresponding measures; therefore, congruent triangles have all corresponding angles of equal measure and all corresponding sides of equal measure. Two triangles are congruent if they have exactly the same shape and size. In other words, if one triangle can be picked up and situated on top of another triangle so that the two triangles coincide, they are said to be congruent.

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DEFINITION Congruent Triangles STUDY TIP Congruent triangles are triangles with equal corresponding angle measures (equal Similar triangles: Exact same shape Congruent triangles: Exact angles) and corresponding equal side lengths. same shape and size

All congruent triangles are also similar triangles, but not all similar triangles are congruent triangles. Trigonometry (as you will see in Section 1.3) relies on the properties of similar 30º triangles. Since similar triangles have the same shape (equal corresponding angles), 2x the sides opposite the corresponding angles must be proportional. √3x Given any 308-608-908 triangle (see figure in the margin), if we let x 5 1 correspond to triangle A and x 5 5 correspond to triangle B, then we would have the following two triangles. 60º x

Triangle B 5 Sides opposite 308: 5 5 5 Triangle A 1 30º

Triangle B 5!3 10 Sides opposite 608: 5 5 5 5 3 Triangle A !3 √ B 30º 2 Triangle B 10 3 Sides opposite 908: 5 5 5 √ A Triangle A 2 60º 60º 1 5

Notice that the sides opposite the corresponding angles are proportional. This means that we will find that all three ratios of each side of triangle B to its corresponding side of triangle A are equal. This proportionality property holds for all similar triangles.

CONDITIONS FOR SIMILAR TRIANGLES One of the following must be verified in order for two triangles to be similar: ba ■■ ■Corresponding angles must have the same measure. or c

■■ ■Corresponding sides must be proportional (ratios must be equal) b' a' a b c 5 5 c' ar br cr

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Let us now use the fact that corresponding sides of similar triangles are proportional to determine lengths of sides of similar triangles.

EXAMPLE 2 Finding Lengths of Sides in Similar Triangles Given that the two triangles are similar, find the length of each of the unknown sides 1b and c2. b 8

6 Solution: Solve for b. 8 b The corresponding sides are proportional. 5 2 5 Multiply by 5. b 5 4152 Solve for b. b 5 20

Solve for c. 8 c The corresponding sides are proportional. 5 2 6 Multiply by 6. c 5 4162 Solve for c. c 5 24 8 20 24 Check that the ratios are equal: 5 5 5 4 2 5 6

▼ ▼ ANSWER YOUR TURN Given that the two triangles are similar, find the length of each a 5 5, b 5 8 of the unknown sides 1a and b2.

24 27

b 9

15 a

Applications Involving Similar Triangles As you have seen, the common ratios associated with similar triangles are very useful. For example, you can quickly estimate the heights of flagpoles, trees, and any other tall objects by measuring their shadows along the ground because similar triangles are formed. Surveyors rely on the properties of similar triangles to determine distances that are difficult to measure.

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EXAMPLE 3 Calculating the Height of a Tree Billy wants to rent a lift to trim his tall trees. However, he must decide which lift he needs: one that will lift him 25 feet or a more expensive lift that will lift him 50 feet. His wife Jeanine hammered a stake into the ground and by measuring found its shadow to be 1.75 feet long and the tree’s shadow to be 19 feet. (Assume both the stake and tree are perpendicular to the ground.) If the stake was standing 3 feet above the ground, how tall is the tree? Which lift should Billy rent?

(NOT TO SCALE) x ft

3 ft

1.75 ft 19 ft

Solution: Rays of sunlight are straight and parallel to each other. Therefore, the rays make the same angle with the tree that they do with the stake. (NOT TO SCALE) x Draw and label the two similar triangles.

19 1.75

The ratios of corresponding sides of similar x 19 5 triangles are equal. 3 1.75 31192 Solve for x. x 5 1.75 Simplify. x < 32.57

The tree is approximately 33 feet tall. Billy should rent the more expensive lift to be safe.

▼ ▼ YOUR TURN Billy’s neighbor decides to do the same thing. He borrows ANSWER Jeanine’s stake and measures the shadows. If the shadow of approximately 38 ft his tree is 15 feet and the shadow of the stake (3 feet above the ground) is 1.2 feet, how tall is Billy’s neighbor’s tree? Round to the nearest foot.

[SECTION 1.2] SUMMARY

Similar triangles (the same shape) have corresponding angles lengths. Similar triangles have the property that the ratios of their with equal measure. Congruent triangles (the same shape and corresponding side lengths are equal. size) are similar triangles that also have equal corresponding side

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[SECTION 1.2] EXERCISES

• SKILLS In Exercises 1–12, find the measure of the indicated angles, using the diagram on the right. 1. C 5 80°, find B. 2. C 5 80°, find E. 3. C 5 80°, find F. 4. C 5 80°, find G. m||n 5. F 5 75°, find B. 6. F 5 75°, find A. 7. G 5 65°, find B. 8. G 5 65°, find A. AB m 9. A 5 18x2° and D 5 19x 2 152°, find the measures of A and D. CD 10. B 5 19x 1 72° and F 5 111x 2 72°, find the measures of B and F. 11. A 5 112x 1 142° and G 5 19x 2 22°, find the measures of A and G. EF GH n 12. C 5 122x 1 32° and E 5 130x 2 52°, find the measures ofC and E. In Exercises 13–18, match the corresponding triangle with the appropriate name. 13. Equilateral triangle 14. Right triangle (nonisosceles) 15. Isosceles triangle (nonright) 16. Acute scalene triangle 17. Obtuse scalene triangle 18. Isosceles right triangle a. b. c. d. e. f.

In Exercises 19–26, calculate the specified lengths, given that the two triangles are similar.

19. a 5 4, c 5 6, d 5 2, f 5 ? 20. a 5 12, b 5 9, e 5 3, d 5 ? 21. d 5 5, e 5 2.5, b 5 7.5, a 5 ? 22. e 5 1.4, f 5 2.6, c 5 3.9, b 5 ? ac 23. d 5 1.1 m, f 5 2.5 m, c 5 26.25 km, a 5 ? 24. e 5 10 cm, f 5 14 cm, c 5 35 m, b 5 ? df 4 5 3 7 1 5 25. b 5 5 in., c 5 2 in., f 5 2, e 5 ? 26. d 5 16 m, e 5 4 m, b 5 4 mm, a 5 ?

b e

• APPLICATIONS 1 27. Height of a Tree. The shadow of a tree measures 144 feet. At century. On a sunny day, if a 2-meter-tall man casts a shadow the same time of day, the shadow of a 4-foot pole measures approximately 5 centimeters (0.05 meters) long, and the 1 12 feet. How tall is the tree? lighthouse approximately a 3-meter shadow, how tall was 28. Height of a Flagpole. The shadow of a flagpole measures this fantastic structure? 15 feet. At the same time of day, the shadow of a stake 2 feet 3 above ground measures 4 foot. How tall is the flagpole? 29. Height of a Lighthouse. The Cape Hatteras Lighthouse at the Outer Banks of North Carolina is the tallest lighthouse in 1 North America. If a 5-foot woman casts a 15-foot shadow and the lighthouse casts a 48-foot shadow, approximately how tall is the Cape Hatteras Lighthouse? 30. Height of a Man. If a 6-foot man casts a 1-foot shadow, how long a shadow will his 4-foot son cast? For Exercises 31 and 32, refer to the following: Although most people know that a list exists of the Seven Wonders of the Ancient World, only a few can name them: the Great Pyra- mid of Giza, the Hanging Gardens of Babylon, the Statue of Zeus at Olympia, the Temple of Artemis at Ephesus, the Mausoleum of Halicarnassus, the Colossus of Rhodes, and the Lighthouse of Alexandria. Mary Evans Picture Library/Alamy 31. Seven Wonders. One of the Seven Wonders of the Ancient World was a lighthouse on the Island of Pharos in Alexan- 32. Seven Wonders. Only one of the great Seven Wonders dria, Egypt. It is the first lighthouse in recorded history and of the Ancient World is still standing—the Great Pyramid was built about 280 bc. It survived for 1500 years until it of Giza. Each of the base sides along the ground measures was completely destroyed by an earthquake in the fourteenth 230 meters. If a 1-meter child casts a 90-centimeter shadow

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at the same time the shadow of the pyramid extends 16 meters along the ground (beyond the base), approximately how tall is the Great Pyramid of Giza?

38. Design. Rita’s supervisor asks that the triangles in Rita’s design (shown in Exercise 37) be isosceles right triangles. If Rita decides to make the hypotenuse of the larger triangle 2 inches long, how long are its legs?

For Exercises 39–42, refer to the following: A collimator is a device used in radiation treatment that narrows beams or waves, causing the waves to be more aligned in a

Dan Breckwoldt/Shutterstock.com specific direction. The use of a collimator facilitates the focusing of radiation to treat an affected region of tissue beneath the skin. In In Exercises 33 and 34, use the drawing below: the figure, ds is the distance from the radiation source to the skin, dt is the distance from the outer layer of skin to the targeted tissue, 2fs is the field size on the skin (diameter of the circular treated skin) and 2fd is the targeted field size at depthd t (the diameter of the targeted tissue at the specified depth beneath the skin surface).

Island Pantry Radiation source

Collimator

In a home remodeling project, your architect gives you plans that 1 have an indicated distance of 2r4s, which measures 4 inch with a ruler on the blueprint. 33. Measurement. How long is the pantry in the kitchen if it ds 11 measures 16 inch with a ruler? 7 34. Measurement. How wide is the island if it measures 16 inch f with a ruler? s dt

35. Camping. The front of Brian’s tent is a triangle that measures fd 5 feet high and 4 feet wide at the base. The tent has one zipper down the middle that is perpendicular to the base of the tent and one zipper that is parallel to the base. With the tent 39. Health/Medicine. Radiation treatment is applied to a field unzipped, the opening and the outline of the tent are equilateral size of 8 centimeters at a depth 2.5 centimeters below the skin triangles as shown below. If the vertical zipper measures surface. If the treatment head is positioned 80 centimeters from 3.5 feet, how wide is the opening at the base? the skin, find the targeted field size to the nearest millimeter. 40. Health/Medicine. Radiation treatment is applied to a field size of 4 centimeters lying at a depth of 3.5 centimeters below the skin surface. If the field size on the skin is required to be 3.8 centimeters, find the distance from the skin that the radiation source must be located to the nearest millimeter. 41. Health/Medicine. Radiation treatment is applied to a field size on the skin of 3.75 centimeters to reach an affected region 36. Camping. Zach sees a tree growing at an angle with the ground. of tissue with field size of 4 centimeters at some depth below If the angle between the tree and the ground is 60°, how far is it the skin. If the treatment head is positioned 60 centimeters from the ground to the tree 15 feet from the base of the tree? from the skin surface, find the desired depth below the skin to 37. Design. Rita is designing a logo for her company’s letterhead. the target area to the nearest millimeter. As part of her design, she is trying to include two similar right 42. Health/Medicine. Radiation treatment is applied to a field triangles within a circle as shown. If the vertical leg of the large size on the skin of 4.15 centimeters to reach an affected area triangle measures 1.6 inches, while the hypotenuse measures lying 4.5 centimeters below the skin surface. If the treatment 2.2 inches, what is the length of the vertical leg of the smaller head is positioned 60 centimeters from the skin surface, find triangle given that its hypotenuse measures 1.2 inches? the field size of the targeted area to the nearest millimeter.

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• CATCH THE MISTAKE In Exercises 43 and 44, explain the mistake that is made. 43. In the similar triangles shown, if A 5 8, B 5 5, and D 5 3, 44. In the similar triangles shown in Exercise 43, if A 5 8, find E. B 5 5, and D 5 3, find F.

Solution: A D AC Set up a ratio of similar triangles. 5 DF B F 8 3 B E Substitute A 5 8, B 5 5, and 5 D 5 3. 5 F Solution: A B Cross multiply. 8F 5 5 3 Set up a ratio of corresponding sides. 5 1 2 E D 15 8 5 Solve for F. F 5 Substitute A 5 8, B 5 5, and D 5 3. 5 8 E 3 Cross multiply. 5E 5 8132 This is incorrect. What mistake was made? 24 Solve for E. E 5 5 This is incorrect. What mistake was made?

• CONCEPTUAL In Exercises 45–52, determine whether each statement is true or false. 45. Two similar triangles must have equal corresponding angles. 49. Alternate interior angles are supplementary. 46. All congruent triangles are similar, but not all similar trian- 50. Vertical angles are congruent. gles are congruent. 51. All isosceles triangles are similar to each other. 47. Two angles in a triangle cannot have measures 82° and 67°. 52. Corresponding sides of similar triangles are congruent. 48. Two equilateral triangles are similar but do not have to be congruent.

• CHALLENGE

53. Find x. Image Do

Ho 1 f 3 Di – f 4 x Do – f 2 f 4 Hi

Di 3 Object 3 6 55. Explain why triangles 1 and 2 are similar triangles and why triangles 3 and 4 are similar triangles. 56. Set up the similar triangle ratios for 54. Find x and y. a. triangles 1 and 2 b. triangles 3 and 4 57. Use the ratios in Exercise 56 to derive the Lens law. 4 y For Exercises 58–60, use the figure below:

5 BF is parallel to CG, CD is congruent to CG, and the measure of x 18 angle A 5 30°. A

EF 20 B

C For Exercises 55–57, refer to the following: G The Lens law relates three quantities: the distance from the object to the lens, Do; the distance from the lens to the image, Di; and the D focal length of the lens, ƒ. 58. Name all similar triangles in the figure. 59. If AB measures 2 centimeters and BC measures 3 centimeters, 1 1 1 1 5 find the measure of EF and EG. Do Di ƒ 60. What is the measure of DG?

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1.3 DEFINITION 1 OF TRIGONOMETRIC FUNCTIONS: RIGHT TRIANGLE RATIOS SKILLS OBJECTIVES CONCEPTUAL OBJECTIVES ■■ Calculate trigonometric ratios of general acute angles. ■■ Understand that right triangle ratios are based on the ■■ Express trigonometric functions in terms of their properties of similar triangles. cofunctions. ■■ Understand that a trigonometric function of an angle is the cofunction of its complementary angle.

1.3.1 Trigonometric Functions: Right Triangle Ratios

The word trigonometry stems from the Greek words trigonon, which means “triangle,” 1.3.1 SKILL and metrein, which means “to measure.” Trigonometry began as a branch of geometry Calculate trigonometric ratios of and was utilized extensively by early Greek mathematicians to determine unknown general acute angles distances. The trigonometric functions were first defined as ratios of side lengths in a right triangle. This is the way we will define them in this section. Since the two angles 1.3.1 CONCEPTUAL besides the right angle in a right triangle have to be acute, a second kind of definition was needed to extend the domain of trigonometric functions to nonacute angles in Understand that right triangle the Cartesian plane (Sections 2.2 and 2.3). Starting in the eighteenth century, broader ratios are based on the properties definitions of the trigonometric functions came into use, under which the functions are of similar triangles. associated with points along the unit circle (Section 3.4). In Section 1.2, we reviewed the fact that triangles with equal corresponding angles also have proportional sides and are called similar triangles. The concept of similar triangles (one of the basic insights in trigonometry) allows us to determine the length of a side of one triangle if we know the length of one side of that triangle and the length of certain sides of a similar triangle. Since the two right triangles to the right have equal angles, they are similar triangles, c and the following ratios hold true: a c' a' a b c 5 5 ar br cr b b'

WORDS MATH a b Start with the first ratio. 5 ar br [ CONCEPT CHECK] Cross multiply. abr 5 arb How many possible ratios of the length of the sides of a right abr arb Divide both sides by bbr. 5 triangle are there for an acute bbr bbr angle in a right triangle? ▼ a ar Simplify. 5 ANSWER 6 b br

b br a ar Similarly, it can be shown that 5 and 5 . c cr c cr

Notice that even though the sizes of the triangles are different, since the corresponding angles are equal, the ratio of the lengths of the two legs of the large triangle is equal to a ar the ratio of the lengths of the legs of the small triangle, or 5 . Similarly, the ratios b br of the lengths of a leg and the hypotenuse of the large triangle and the corresponding b br a ar leg and hypotenuse of the small triangle are also equal; that is, 5 and 5 . c cr c cr

Young_Trigonometry_6161_ch01_pp002-035.indd 25 15/12/16 3:00 PM 26 CHAPTER 1 Right Triangle Trigonometry

For any right triangle, there are six possible ratios of the length of the sides that can be calculated for each acute angle u: b a b c c a c b c c a b a b These ratios are referred to as trigonometric ratios or trigonometric functions, since θ they depend on u, and each is given a name: a

FUNCTION NAME ABBREVIATION WORDS MATH

Sine sin The sine of u sin u

Cosine cos The cosine of u cos u

Tangent tan The tangent of u tan u

Cosecant csc The cosecant of u csc u

Secant sec The secant of u sec u

Cotangent cot The cotangent of u cot u

Sine, cosine, tangent, cotangent, secant, and cosecant are names given to specific ratios of lengths of sides of a right triangle. These are the six trigonometric functions.

DEFINITION Trigonometric Functions Let u be an acute angle in a right triangle; then b a b sin u 5 cos u 5 tan u 5 c c a c c c a b csc u 5 sec u 5 cot u 5 b a b θ In this right triangle, we say that: a ■■ Side c is the hypotenuse. ■■ Side b is the side (leg) opposite angle u. ■■ Side a is the side (leg) adjacent to angle u.

b b a sin u c b 5 5 tan u Also notice that since sin u 5 c and cos u 5 c, then 5 a a . cos u c

The three main trigonometric functions should be learned in terms of the ratios.

opposite adjacent opposite sin u 5 cos u 5 tan u 5 hypotenuse hypotenuse adjacent

The remaining three trigonometric functions can be derived from sin u, cos u, and 1 STUDY TIP tan u using the reciprocal identities. Recall that the reciprocal of x is for x 2 0. x You only need to memorize the three main trigonometric ratios: sin u, cos u, and tan u. The remaining three can always be RECIPROCAL IDENTITIES calculated as reciprocals of the main three for an acute angle u 1 1 1 by remembering the reciprocal csc u 5 sec u 5 cot u 5 identities. sin u cos u tan u

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It is important to note that the reciprocal identities only hold for values of u that do STUDY TIP not make the denominator equal to zero (i.e., when sin u, cos u, or tan u are not equal to 0). Using this terminology, we have an alternative definition that is easier to remember. Trigonometric functions are functions of a specified angle. Always specify the angle. Sin

alone means nothing. Sin u DEFINITION Trigonometric Functions (Alternate Form) specifies the angle dependency. Let u be an acute angle in a right triangle; then The same is true for the other five trigonometric functions. opposite sin u 5 hypotenuse adjacent c Hypotenuse b cos u 5 hypotenuse Opposite opposite θ tan u 5 adjacent a STUDY TIP Adjacent SOHCAHTOA is an acrostic that and their reciprocals is often used to remember the right triangle definitions of sine, 1 1 1 cosine, and tangent. csc u 5 sec u 5 cot u 5 sin u cos u tan u opposite SOH: sin u 5 hypotenuse

Notice that the tangent function can also be written as a quotient of the sine function adjacent CAH: cos u 5 and the cosine function: hypotenuse opposite opposite

TOA: tan u 5 sin u hypotenuse opposite adjacent tan u 5 5 5 cos u adjacent adjacent hypotenuse

cos u and similarly, cot u 5 . These relationships are called the quotient identities. sin u

EXAMPLE 1 Finding Trigonometric Function Values of a General Angle u in a Right Triangle For the given right triangle, calculate the values of

sin u, tan u, and csc u.

θ Solution: 3

STEP 1 Find the length of the hypotenuse.

x 4

θ 3

Apply the Pythagorean theorem. 42 1 32 5 x2 16 1 9 5 x2 x2 5 25 x 5 65 Lengths of sides can be only positive. x 5 5

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STEP 2 Label the sides of the triangle:

■■ with numbers representing lengths. ■■ as hypotenuse or as opposite or adjacent 5 4 Hypotenuse with respect to u. Opposite

θ 3 Adjacent STEP 3 Set up the trigonometric functions as ratios. opposite 4 Sine is Opposite over Hypotenuse (SOH). sin u 5 5 hypotenuse 5

opposite 4 Tangent is Opposite over Adjacent (TOA). tan u 5 5 adjacent 3

1 1 5 Cosecant is the reciprocal of sine. csc u 5 5 5 sin u 4 4 ▼ ANSWER 5 3 5 cos u 5 5, sec u 5 3, 3 ▼ and cot u 5 4 YOUR TURN For the triangle in Example 1, calculate the values of cos u, sec u,

and cot u.

EXAMPLE 2 Finding Trigonometric Function Values for a General Angle u in a Right Triangle For the given right triangle, calculate

cos u, tan u, and sec u. 7

65 Solution: √

STEP 1 Find the length of the unknown leg.

x 7

√65

2 Apply the Pythagorean theorem. x2 1 72 5 A!65B x2 1 49 5 65 x2 5 16 x 5 64 Lengths of sides can be only positive. x 5 4

STEP 2 Label the sides of the triangle: Opposite Adjacent ■■ 7 with numbers representing lengths. 4 ■■ as hypotenuse or as opposite or adjacent θ with respect to u. √65 Hypotenuse

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STEP 3 Set up the trigonometric functions as ratios. STUDY TIP Because we rationalize adjacent 4 expressions containing radicals Cosine is Adjacent over Hypotenuse (CAH). cos u 5 5 in the denominator, sometimes hypotenuse !65 reciprocal ratios may not always look like reciprocal fractions: opposite 7 4 65 65 Tangent is Opposite over Adjacent (TOA). tan u 5 5 " " cos u 5 ; sec u 5 . adjacent 4 65 4 Secant is the reciprocal of cosine. 1 1 !65 sec u 5 5 5 cos u 4 4 !65 4 Expressions that contain a radical in the denominator like can be rationalized !65 by multiplying both the numerator and the denominator by the radical, !65.

4 !65 4!65 # a b 5 !65 !65 65 g 1 4!65 In this example, the cosine function can now be written as cos u 5 . 65

▼ ▼ YOUR TURN For the triangle in Example 2, calculate sin u, csc u, and cot u. ANSWER

7!65 sin u 5 , 65 !65 4 csc u 5 , and cot u 5 7 7 1.3.2 Cofunctions

Notice the co in cosine, cosecant, and cotangent functions. These cofunctions are 1.3.2 SKILL based on the relationship of complementary angles. Let us look at a right triangle with Express trigonometric functions labeled sides and angles. in terms of their cofunctions.

1.3.2 CONCEPTUAL α opposite of b b sin b 5 5 Understand that a trigonometric c hypotenuse c function of an angle is the b sin b 5 cos a adjacent to a b cofunction of its complementary cos a 5 5 angle. β hypotenuse c ∂ a Recall that the sum of the measures of the three angles in a triangle is 180°. In a right triangle, one angle is 90°; therefore, the two acute angles are complementary angles (the measures sum to 90°2. You can see in the triangle above that b and a are complementary angles. In other words, the sine of an angle is the same as the cosine of the complement of that angle. This is true for all trigonometric cofunction pairs.

CONCEPT CHECK COFUNCTION THEOREM [ ] TRUE OR FALSE: A trigonometric function of an angle is always equal to the cofunction of the sin u 5 cos190° 2 u2 complement of the angle. If a 1 b 5 90°, then ▼

sin b 5 cos a ANSWER True

sec b 5 csc a

tan b 5 cot a

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COFUNCTION IDENTITIES

sin u 5 cos190° 2 u2 cos u 5 sin190° 2 u2

tan u 5 cot190° 2 u2 cot u 5 tan190° 2 u2

sec u 5 csc190° 2 u2 csc u 5 sec190° 2 u2

90º – θ c b

θ a

EXAMPLE 3 Writing Trigonometric Function Values in Terms of Their Cofunctions Write each function or function value in terms of its cofunction.

a. sin 30° b. tan x c. csc 40° Solution (a):

Cosine is the cofunction of sine. sin u 5 cos190° 2 u2

Substitute u 5 30°. sin 30° 5 cos190° 2 30°2

Simplify. sin 30° 5 cos 60° Solution (b):

Cotangent is the cofunction of tangent. tan u 5 cot190° 2 u2

Substitute u 5 x. tan x 5 cot190° 2 x2 Solution (c):

Secant is the cofunction of cosecant. csc u 5 sec190° 2 u2

Substitute u 5 40°. csc 40° 5 sec190° 2 40°2

Simplify. csc 40° 5 sec 50°

▼ ▼ ANSWER YOUR TURN Write each function or function value in terms of its cofunction.

a. sin 45° b. sec190° 2 y2 a. cos 45° b. csc y

[SECTION 1.3] SUMMARY

In this section, we have defined trigonometric functions as ratios opposite adjacent opposite sin u 5 cos u 5 tan u 5 of the lengths of sides of right triangles. This approach is called hypotenuse hypotenuse adjacent right triangle trigonometry. This is the first of three definitions of trigonometric functions (others will follow in Chapters 2 and It is important to remember that “adjacent” and “opposite” are 3). We now can find trigonometric functions of an acute angle with respect to one of the acute angles we are considering. We by taking ratios of the three sides of a right triangle: adjacent, learned that the trigonometric functions of an angle are equal to opposite, and hypotenuse. the cofunctions of the complement to the angle.

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[SECTION 1.3] EXERCISES

• SKILLS In Exercises 1–6, refer to the triangle in the drawing to find the indicated trigonometric function values.

1. sin u 2. cos u 3. csc u 4. sec u 5. tan u 6. cot u 10 8

θ 6 In Exercises 7–12, refer to the triangle in the drawing to find the indicated trigonometric function values. Rationalize any denominators containing radicals that you encounter in the answers.

7. cos u 8. sin u 9. sec u 10. csc u 11. tan u 12. cot u

θ 1 In Exercises 13–18, refer to the triangle in the drawing to find the indicated trigonometric function values. Rationalize any denominators containing radicals that you encounter in the answers.

13. sin u 14. cos u 15. tan u 16. csc u 17. sec u 18. cot u 5

√34 In Exercises 19–24, use the cofunction identities to fill in the blanks.

19. sin 60° 5 cos______20. sin 45° 5 cos______

21. cos x 5 sin______22. cot A 5 tan______

23. csc 30° 5 sec______24. sec B 5 csc______

In Exercises 25–30, write the trigonometric function values in terms of its cofunction. 25. sin1x 1 y2 26. sin160° 2 x2 27. cos120° 1 A2 28. cos1A 1 B2 29. cot145° 2 x2 30. sec130° 2 u2

• APPLICATIONS

For Exercises 31 and 32, consider the following scenario: A man lives in a house that borders a pasture. He decides to go to 3 4 31. Shortcut. If sin u 5 5 and cos u 5 5 and if he drove his car the grocery store to get some milk. He is trying to decide whether along the streets, it would be 14 miles round trip. How far to drive along the roads in his car or take his all terrain vehicle would he have to go on his ATV round trip? Round your (ATV) across the pasture. His car drives faster than the ATV, but answer to the nearest mile. the distance the ATV would travel is less than the distance he 32. Shortcut. If tan u 5 1 and if he drove his car along the would travel in his car. streets, it would be 200 yards round trip. How far would he have to go on his ATV round trip? Round your answer to the nearest yard. 33. Roofing. Bob is told that the pitch on the roof of his garage is 5-12, meaning that for every 5 feet the roof increases vertically, it increases 12 feet horizontally. If u is defined as the angle at the corner of the roof formed by the pitch of the

roof and a horizontal line, what is sin u? 34. Roofing. Bob’s roof has a 5-12 pitch, while his neighbor’s θ roof has a 7-12 pitch. With u defined as the angle formed at the corner of the roof by the pitch of the roof and a horizontal

line, whose roof has a larger value for cos u? Explain.

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35. Automotive Design. The angle u formed by a car’s windshield Top view Side view and dashboard is such that tan u 5 !3. What is the measure of angle u? Drop Blood Direction of drop travel

w y θ indshield Blood W l splatter Dashboard 41. Forensic Science. If a drop of blood found at a crime scene has a width of 6 millimeters and a length of 12 millimeters, 36. Automotive Design. Consider the information related to find the angleu that represents the directionality. the interior of the car given in Exercise 35. If the vertical distance y between the top of the windshield and the horizontal 42. Forensic Science. If a drop of blood found at a crime scene plane of the dashboard is 3 feet, what is the length of the has a width of !2 millimeters and a length of 2 millimeters, windshield? find the angle u that represents the directionality. 37. Bookshelves. Juan is building a bookcase. To give each shelf For Exercises 43 and 44, refer to the following: extra strength, he is planning to add a triangle-shaped brace on each end. If each brace is a right triangle as shown below, The monthly profits of PizzaRia are a function of sales, that is,p (s). 3 A financial analysis has determined that the saless in thousands of such that cos A 5 , finds in B. 5 dollars of PizzaRia are also related to monthly profitsp in thousands of dollars by the relationship p tan u 5 for 0 # s # 55 and 0 # p # 45 s A

B Prots (p)

p(s)

38. Bookshelves. If the height of the brace shown in Exercise 37 is 8 inches, find the width of the brace. 39. Hiking. Raja and Ariel are planning to hike to the top of a p θ Sales (s) hill. If u is the angle formed by the hill and the ground as s shown below, such that sec u 5 1.75, finds in u.

Based on sales and profits, it can be determined that the domain for angle u is

0° # u # 40°

The ratio tan u represents the slope of the hypotenuse of the right triangle formed by sales s and profit p (see figure above). The angle u can be interpreted as a measure of the relative size of s to p. The larger the angle u is, the greater profit is relative to sales, 40. Hiking. Having recently had knee surgery, Lila is under doctor’s and conversely, the smaller the angle u is, the smaller profit is orders not to hike anything steeper than a 45° incline. If u is the relative to sales. angle formed by the hill and the ground such that tan u 5 1.2, is 43. Business. If PizzaRia’s monthly sales are $25,000 and the hill too steep for Lila to hike? Explain. monthly profits are $10,000, find

For Exercises 41 and 42, refer to the following: a. tan u b. cot u When traveling through air, a spherical drop of blood with diam- 44. Business. If PizzaRia’s monthly sales are s and monthly profits eter d maintains its spherical shape until hitting a flat surface. The are p: drop of blood’s direction of travel dictates the directionality of a. Determine the hypotenuse in terms of s and p. the blood splatter on the surface. For this reason, the diameter of b. Determine a formula for cos u in terms of s and p. the blood drop is equal to the width of the blood splatter on the surface. The angle at which a spherical drop of blood is deposited on a surface, called angle of impact, is related to the width w and w the length l of the splatter by sin u 5 . l

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• CATCH THE MISTAKE In Exercises 45–48, explain the mistake that is made.

47. Calculate sec x.

y Solution: 5 4 opposite Write the sine ratio. sin x 5 hypotenuse x The opposite side is 4; the 4 sin x 5 3 hypotenuse side is 5. 5

45. Calculate sin y. 1 Write secant as the reciprocal of sine. sec x 5 sin x Solution: 1 5 opposite Simplify. sec x 5 5 Write the sine ratio. sin y 5 4/5 4 hypotenuse This is incorrect. What mistake was made? The opposite side is 4; the 4 sin y 5 hypotenuse side is 5. 5 48. Calculate csc y.

This is incorrect. What mistake was made? Solution:

46. Calculate tan x. adjacent Write the cosine ratio. cos y 5 hypotenuse Solution: The adjacent side is 4; the 4 cos y 5 adjacent hypotenuse side is 5. 5 Write the tangent ratio. tan x 5 opposite Write cosecant as the reciprocal 1 The adjacent side is 3; the 3 csc y 5 of cosine. cos y tan x 5 opposite side is 4. 4 1 5 Simplify. csc y 5 5 This is incorrect. What mistake was made? 4/5 4 This is incorrect. What mistake was made?

• CONCEPTUAL

In Exercises 49–52, determine whether each statement is true or false.

49. sin 45° 5 cos 45° 50. sin 60° 5 cos 30° 51. sec 60° 5 csc 30° 52. cot 45° 5 tan 45° In Exercises 53–58, use the special triangles (308-608-908 and 458-458-908) shown below:

53. Calculate sin 30° and cos 30°. 56. Calculate sin 45°, cos 45°, and tan 45°. 30º 54. Calculate sin 60° and cos 60°. 57. Calculate sec 45° and csc 45°.

55. Calculate tan 30° and tan 60°. 58. Calculate tan 60° and cot 30°. 2x 45º √3x x √2x

45º 60º x x

• CHALLENGE

59. What values can sin u and cos u take on? 63. As u increases from 0° to 90°, how does the cofunction of

sin u change? 60. What values can tan u and cot u take on? 1 64. As u increases from 0° to 90°, how does csc u 5 61. What values can sec u and csc u take on? change? sin u 62. As u increases from 0° to 90°, how does sin u change?

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• TECHNOLOGY

65. Calculate sec 70° the following two ways: 67. Calculate cot154.9°2 the following two ways:

a. Find cos 70° to three decimal places and then divide 1 by a. Find tan154.9°2 to three decimal places and then divide that number. Write that number to five decimal places. 1 by that number. Write that number to five decimal

b. First findc os 70° and then find its reciprocal. Round the places. result to five decimal places. b. First findt an154.9°2 and then find its reciprocal. Round

66. Calculate csc 40° the following two ways: the result to five decimal places.

a. Find sin 40°, write that down (round to three decimal 68. Calculate sec118.682 the following two ways: places), and then divide 1 by that number. Write this last a. Find cos118.682 to three decimal places and then divide result to five decimal places. 1 by that number. Write that number to five decimal

b. First finds in 40° and then find its reciprocal. Round the places. result to five decimal places. b. First findc os118.6°2 and then find its reciprocal. Round the result to five decimal places.

1.4 EVALUATING TRIGONOMETRIC FUNCTIONS: EXACTLY AND WITH CALCULATORS SKILLS OBJECTIVES CONCEPTUAL OBJECTIVES ■■ Evaluate trigonometric functions exactly for special ■■ Learn the trigonometric functions as ratios of sides of angles. a right triangle. ■■ Evaluate (approximate) trigonometric functions using a ■■ Understand the difference between evaluating trigonometric calculator. functions exactly and using a calculator. ■■ Represent partial degrees in either decimal degrees (DD) ■■ Understand that decimal degrees (DD) and degrees- or degrees-minutes-seconds (DMS). minutes-seconds (DMS) are both ways of measuring parts of a degree and be able to recognize equivalent forms.

1.4.1 Evaluating Trigonometric Functions Exactly for Special Angle Measures: 308, 458, and 608

1.4.1 SKILL We now turn our attention to evaluating trigonometric functions for known angles. In Evaluate trigonometric functions this section, we will distinguish between evaluating a trigonometric function exactly and exactly for special angles. approximating the value of a trigonometric function with a calculator. Throughout this text instructions will specify which is desired (exact or approximate), and the proper notation, 5 or <, will be used. 1.4.1 CONCEPTUAL Three special acute angles are very important in trigonometry: 30°, 45°, and 60°. Learn the trigonometric In Section 1.1, we discussed two important triangles: 308-608-908 and 458-458-908. functions as ratios of sides Recall the relationships between the lengths of the sides of these two right triangles. of a right triangle.

30º

2x 45º √3x x √2x

45º 60º x x

We can combine these relationships with the trigonometric ratios developed in Section 1.3 to evaluate the trigonometric functions for the special angle measures of 30°, 45°, and 60°.

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EXAMPLE 1 Evaluating the Trigonometric CONCEPT CHECK Functions Exactly for 308 [ ] TRUE OR FALSE If you learn Evaluate the six trigonometric functions for an angle that measures 30°. the values of sine and cosine for Solution: the special angles 30°, 45°, and 60°, then you can calculate the Label the sides (opposite, adjacent, and other values (secant, cosecant, hypotenuse) of the 308-608-908 right tangent, and cotangent) using the triangle with respect to the 30° angle. 30º reciprocal and quotient identities. ▼ 2x √3x ANSWER True Adjacent Hypotenuse

x Opposite Use the right triangle ratio definitions of the sine, cosine, and tangent functions. opposite x 1 sin 30° 5 5 5 hypotenuse 2x 2 STUDY TIP

!3 adjacent !3x !3 cos 30° 5 is exact, whereas if cos 30° 5 5 5 2 hypotenuse 2x 2 we evaluate with a calculator, we get an approximation: opposite x 1 1 !3 !3 tan 30° 5 5 5 5 # 5 cos 30° < 0.8660 adjacent !3x !3 !3 !3 3 Use the reciprocal identities to obtain values for the cosecant, secant, and cotangent functions.

1 1 csc 30° 5 5 5 2 sin 30° 1 2 1 1 2 2 !3 2!3 sec 30° 5 5 5 5 # 5 cos 30° !3 !3 !3 !3 3 2 1 1 3 3 !3 cot 30° 5 5 5 5 # 5 !3 tan 30° !3 !3 !3 !3 3 The six trigonometric functions evaluated for an angle measuring 30° are

1 !3 !3 sin 30° 5 cos 30° 5 tan 30° 5 2 2 3

2!3 csc 30° 5 2 sec 30° 5 cot 30° 5 !3 3 ▼ ▼ YOUR TURN Evaluate the six trigonometric functions for an angle that measures 60°. ANSWER

!3 2!3 sin 60° 5 csc 60° 5 In comparing our answers in Example 1 and in Your Turn, we see that the following 2 3 1 cofunction relationships are true: cos 60° 5 sec 60° 5 2 2

sin 30° 5 cos 60° sec 30° 5 csc 60° tan 30° 5 cot 60° !3 tan 60° 5 !3 cot 60° 5 3 sin 60° 5 cos 30° sec 60° 5 csc 30° tan 60° 5 cot 30° which is expected, since 30° and 60° are complementary angles.

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EXAMPLE 2 Evaluating the Trigonometric Functions Exactly for 458 Evaluate the six trigonometric functions for an angle that measures 45°. Solution: Label the sides of the 458-458-908 right triangle with respect to one of the 45° angles.

Hypotenuse x √2x Opposite

45º x Adjacent

Use the right triangle ratio definitions of the sine, cosine, and tangent functions.

opposite x 1 1 !2 !2 sin 45° 5 5 5 5 # 5 hypotenuse !2x !2 !2 !2 2

adjacent x 1 1 !2 !2 cos 45° 5 5 5 5 # 5 hypotenuse !2x !2 !2 !2 2

opposite x tan 45° 5 5 5 1 adjacent x

STUDY TIP Use the reciprocal identities to obtain the values of the cosecant, secant, and cotangent functions. !2 sin 458 5 is exact, whereas if 2 we evaluate with a calculator, we 1 1 2 2 !2 csc 45° 5 5 5 5 # 5 !2 get an approximation: sin 45° !2 !2 !2 !2

sin 45° < 0.7071 2

1 1 2 2 !2 sec 45° 5 5 5 5 # 5 !2 cos 45° !2 !2 !2 !2 2

1 1 cot 45° 5 5 5 1 tan 45° 1

The six trigonometric functions evaluated for an angle measuring 45° are

!2 !2 sin 45° 5 cos 45° 5 tan 45° 5 1 2 2

csc 45° 5 !2 sec 45° 5 !2 cot 45° 5 1

We see that the following cofunction relationships are indeed true:

sin 45° 5 cos 45° sec 45° 5 csc 45° tan 45° 5 cot 45° which is expected, since 45° and 45° are complementary angles.

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The trigonometric function values for the three special angle measures 1308, 458, STUDY TIP and 6082 are summarized in the following table: If you memorize the values for sine and cosine for the angles TRIGONOMETRIC FUNCTION VALUES FOR SPECIAL ANGLES (308, 458, AND 608) given in the table, then the other trigonometric function values in u SIN u COS u TAN u COT u SEC u CSC u the table can be found using the quotient and reciprocal identities. 1 30° !3 !3 !3 2!3 2 2 2 3 3

45° !2 !2 1 1 !2 !2 2 2

1 60° !3 !3 !3 2 2!3 STUDY TIP 2 2 3 3 SOHCAHTOA: • Sine is Opposite over Hypotenuse It is important to learn the special values in red for the sine and cosine functions. • Cosine is Adjacent over All other values in the table can be found through reciprocals or quotients of these Hypotenuse two functions. Remember that the tangent function is the ratio of the sine to cosine • Tangent is Opposite over functions. Adjacent opposite

opposite adjacent sin u hypotenuse opposite sin u 5 cos u 5 tan u 5 5 5 hypotenuse hypotenuse cos u adjacent adjacent hypotenuse

1.4.2 Using Calculators to Evaluate (Approximate) Trigonometric Function Values

We will now turn our attention to using calculators to evaluate trigonometric func- 1.4.2 SKILL tions, which sometimes results in an approximation. Scientific and graphing calcula- Evaluate (approximate) trigono- tors have buttons for the sine (sin), cosine (cos), and tangent (tan) functions. Let us metric functions using a calculator. start with what we already know and confirm it with our calculators.

1.4.2 CONCEPTUAL EXAMPLE 3 Evaluating Trigonometric Functions with a Calculator Understand the difference between Use a calculator to find the values of evaluating trigonometric functions exactly and using a calculator. a. cos 30° b. sin 75° c. tan 67° d. sec 52° Round your answers to four decimal places. Solution: STUDY TIP a. cos 30° < 0.866025403 < 0.8660 Make sure your calculator is set in b. sin 75° < 0.965925826 < 0.9659 degrees (DEG) mode.

c. tan 67° < 2.355852366 < 2.3559

1 CONCEPT CHECK d. sec 52° 5 < 1.624269245 < 1.6243 [ ] cos 52° TRUE OR FALSE sin(458) = 0.07071 !3 ▼ Note: We know cos 30° 5 < 0.8660. 2 ANSWER False. That is an approximation. The exact value is !2.

▼ ▼ YOUR TURN Use a calculator to find the values of ANSWER a. 0.9272 b. 6.3138 a. cos 22° b. tan 81° c. csc 37° d. sin 45° 1.6616 0.7071 Round your answers to four decimal places. c. d.

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1.4.3 Representing Partial Degrees: DD or DMS 1.4.3 SKILL Represent partial degrees in Since one revolution is equal to 360°, an angle of 1° seems very small. However, there either decimal degrees (DD) or are times when we want to break down a degree even further. degrees-minutes-seconds (DMS). For example, if we are off even one-thousandth of a degree in pointing our antenna toward a geostationary satellite, we won’t receive a signal. That is because the distance the signal travels (35,000 kilometers) is so much larger than the size of the satellite 1.4.3 CONCEPTUAL (5 meters). Understand that decimal degrees (DD) and degrees-minutes- seconds (DMS) are both ways of measuring parts of a degree and Satellite being able to recognize equivalent forms. Less than 35,000 km 1/1000 of a degree

There are two traditional ways of representing part of a degree: degrees-minutes- seconds (DMS) and decimal degrees (DD). Most calculators allow you to enter angles in either DD or DMS format, and some even make the conversion between them. [ CONCEPT CHECK] However, we will illustrate a manual conversion technique. Fill in the blanks: A8 B9 and C 0 can The degrees-minutes-seconds way of breaking down degrees is similar to how x y be expressed as A 1 60 1 3600°. we break down time in hours-minutes-seconds. There are 60 minutes in an hour ▼ and 60 seconds in a minute, which results in an hour being broken down into 3,600 ANSWER x 5 B and y 5 C seconds. Similarly, we can think of degrees like hours. We can divide 1° into 60 equal parts. Each part is called a minute and is denoted 1r. One minute is therefore 1 60 of a degree.

1 ° 1r 5 a b or 60r 5 1° 60

We can then break down each minute into 60 seconds. Therefore, 1 second, denoted 1 1 1s, is 60 of a minute or 3600 of a degree.

1 r 1 ° 1s 5 a b 5 a b or 60s 5 1r 60 3600

The following examples represent how DMS expressions are stated:

WORDS MATH 22 degrees, 5 minutes 22° 5r

49 degrees, 21 minutes, 17 seconds 49° 21r 17s

We add and subtract fractional values in DMS form similar to how we add and subtract time. For example, if Carol and Donna both run the Boston Marathon and Carol crosses the finish line in 5 hours, 42 minutes, and 10 seconds and Donna crosses the finish line in 6 hours, 55 minutes, and 22 seconds, how much time elapsed between the two women crossing the finish line? The answer is 1 hour, 13 minutes, and 12 seconds. We combine hours with hours, minutes with minutes, and seconds with seconds.

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EXAMPLE 4 Adding Degrees in DMS Form

Add 27° 5r 17s and 35° 17r 52s. Solution: Align degrees with degrees, minutes with minutes, 27° 5r 17s and seconds with seconds. 1 35° 17r 52s Add degrees, minutes, and seconds, respectively. 62° 22r 69s f Note that 69 seconds is equal to 1 minute, 9 seconds 19 1 90 and simplify. 62° 23r 9s

▼ ▼ YOUR TURN Add 35° 21r 42s and 7° 5r 30s. ANSWER 42° 27r 12s

EXAMPLE 5 Subtracting Degrees in DMS Form Subtract 15° 28r from 90°. Solution: Align degrees with degrees and minutes with minutes. 90° 0r 2 15° 28r Borrow 1° from 90° and write it as 60 minutes. 89° 60r 2 15° 28r Subtract degrees and minutes, respectively. 74° 32r

▼ ▼ YOUR TURN Subtract 23°8r from 90°. ANSWER 66° 52r

An alternate way of representing parts of degrees is with decimal degrees. For example, 33.4° and 91.725° are measures of angles in decimal degrees. To convert from DD to DMS, multiply the decimal part once by 60 to get the minutes and—if necessary— multiply the resulting decimal part by 60 to get the seconds. A similar two-stage reverse process is necessary for the opposite conversion of DMS to DD.

EXAMPLE 6 Converting from Degrees-Minutes-Seconds to Decimal Degrees Convert 17°39r22s to decimal degrees. Round to the nearest thousandth. Solution: Write the number of minutes in 39 ° 1 ° 399 5 a b 5 0.65° decimal form, where 1r 5 1602 . 60

Write the number of seconds in 22 ° 1 ° 220 5 a b < 0.0061° decimal form, where 1s 5 136002 . 3600

Write the expression as a sum. 17°39r22s < 17° 1 0.65° 1 0.0061° Add and round to the nearest thousandth. 17°39r22s < 17.656°

▼ ▼ YOUR TURN Convert 62° 8r 15s to decimal degrees. Round to the nearest ANSWER thousandth. 62.138°

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EXAMPLE 7 Converting from Decimal Degrees to Degrees-Minutes-Seconds Convert 29.538° to degrees, minutes, and seconds. Round to the nearest second. Solution: Write as a sum. 29.538° 5 29° 1 0.538° To find the number of minutes, multiply the decimal part by 60, 609 29.538° 5 29° 1 0.538° a b since 1° 5 609. 1° Simplify. 29.538° 5 29° 1 32.28r Write as a sum. 29.538° 5 29° 1 32r 1 0.28r To find the number of seconds, 600 multiply the decimal by 60, 29.538° 5 29° 1 32r 1 0.28ra b 19 since 19 5 600.

Simplify. 29.538° 5 29° 32r 16.8s

Round to the nearest second. 29.538° 5 29° 32r 17s

▼ ▼ ANSWER YOUR TURN Convert 35.426° to degrees, minutes, and seconds. Round to the

35° 25r 34s nearest second.

In this text, we will primarily use decimal degrees for angle measure, and we will also use decimal approximations for trigonometric function values. A common question that arises is how to round the decimals. There is a difference between specifying that a number be rounded to a particular decimal place and specifying rounding to a certain number of significant digits. More discussion on that topic will follow in the next section when solving right triangles. For now, we will round angle measures to the nearest minute in DMS or the nearest hundredth in DD, and we will round trigonometric function values to four decimal places for convenience.

EXAMPLE 8 Evaluating Trigonometric Functions with Calculators Evaluate the following trigonometric functions for the specified angle measurements. Round your answers to four decimal places.

a. sin118° 9r2 b. sec129.524°2 Solution (a): 9 ° Write 18° 9r in decimal degrees. 18° 9r 5 18° 1 a b 60 5 18° 1 0.15° 5 18.15° Use a calculator to evaluate the sine function. sin118.15°2 5 0.311505793 Round to four decimal places. < 0.3115 Solution (b): 1 Write secant as the reciprocal of cosine. sec129.524°2 5 cos129.524°2 Use a calculator to evaluate the expression. sec129.524°2 < 1.149227998 Round to four decimal places. sec129.524°2 < 1.1492

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[SECTION 1.4] SUMMARY

In this section, you have learned the exact values of the sine, u u u u cosine, and tangent functions for the special angle measures: 30°, SIN COS TAN 45°, and 60°. 1 !3 !3 308 The values for each of the other trigonometric functions can 2 2 3 be determined through reciprocal properties. Calculators can be used to approximate trigonometric values of any acute angle. !2 !2 458 1 Degrees can be broken down into smaller parts using one of two 2 2 systems: decimal degrees and degrees-minutes-seconds. !3 1 608 "3 2 2

[SECTION 1.4] EXERCISES

• SKILLS In Exercises 1–6, label each trigonometric function value with its correct value in (a), (b), and (c). 1 !3 !2 a. b. c. 2 2 2

1. sin 30° 2. sin 60° 3. cos 30° 4. cos 60° 5. sin 45° 6. cos 45°

sin u For Exercises 7–9, use the results in Exercises 1–6 and the trigonometric quotient identity, tan u 5 , to calculate the following values. cos u

7. tan 30° 8. tan 45° 9. tan 60° 1 1 For Exercises 10–18, use the results in Exercises 1–9 and the reciprocal identities, csc u 5 , sec u 5 , and sin u cos u 1 cot u 5 , to calculate the following values. tan u

10. csc 30° 11. sec 30° 12. cot 30° 13. csc 60° 14. sec 60° 15. cot 60°

16. csc 45° 17. sec 45° 18. cot 45°

In Exercises 19–30, use a calculator to evaluate the trigonometric functions for the indicated values. Round your answers to four decimal places.

19. sin 37° 20. sin117.8°2 21. cos 82° 22. cos121.9°2 23. tan 54° 24. tan143.2°2

25. sec 8° 26. sec 75° 27. csc 89° 28. csc 51° 29. cot 55° 30. cot 29°

In Exercises 31–38, perform the indicated operations using the following angles:

/A 5 5° 17r 29s /B 5 63° 28r 35s /C 5 16° 11r 30s

31. /A 1 /B 32. /A 1 /C 33. /B 1 /C 34. /B 2 /A 35. /B 2 /C 36. /C 2 /A 37. 90° 2 /A 38. 90° 2 /B

In Exercises 39–46, convert from degrees-minutes-seconds to decimal degrees. Round to the nearest hundredth if only minutes are given and to the nearest thousandth if seconds are given.

39. 33° 20r 40. 89° 45r 41. 59° 27r 42. 72° 13r

43. 27° 45r 15s 44. 36° 5r 30s 45. 42° 28r 12s 46. 63° 10r 9s

In Exercises 47–54, convert from decimal degrees to degrees-minutes-seconds. In Exercises 47–50, round to the nearest minute. In Exercises 51–54, round to the nearest second. 47. 15.75° 48. 15.50° 49. 22.35° 50. 80.47° 51. 30.175° 52. 25.258° 53. 77.535° 54. 5.995°

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In Exercises 55–60, use a calculator to evaluate the trigonometric functions for the indicated values. Round your answers to four decimal places.

55. sin110° 25r2 56. cos175° 13r2 57. tan122° 15r2 58. sec168° 22r2 59. csc128° 25r 35s2 60. sec150° 20r 19s2

• APPLICATIONS

For Exercises 61 and 62, refer to the following: X-ray crystallography is a method of determining the arrangement of atoms within a crystal. This method has revealed the structure Light ray and functioning of many biological molecules including vitamins, Incident drugs, proteins, and nucleic acids (including DNA). The structure angle ni of a crystal can be determined experimentally using Bragg’s law: θiº Surface

nl 5 2d sin u Refractive angle nr where l is the wavelength of the x-ray (measured in angstroms), d is θrº the distance between atomic planes (measured in angstroms), u is the angle of reflection (in degrees), and n is the order of Bragg reflection (a positive integer).

61. Physics/Life Sciences. A diffractometer was used to make Calculate the index of refraction nr of the indicated refractive a diffraction pattern for a protein crystal from which it was medium given the following assumptions. Round answers to three determined experimentally that x-rays of wavelength 1.54 decimal places. angstroms produced an angle of reflection of 458 corresponding ■ The incident medium is air. to a Bragg reflection of order 1. Find the distance between ■ Air has an index of refraction of n 5 1.00. atomic planes for the protein crystal to the nearest hundredth i ■ of an angstrom. The incidence angle is ui 5 30°.

62. Physics/Life Sciences. A diffractometer was used to make a 63. Optics. Diamond, ur 5 12° diffraction pattern for a salt crystal from which it was determined 64. Optics. Emerald, ur 5 18.5° experimentally that x-rays of wavelength 1.67 angstroms 65. Optics. Water, u 5 22° produced an angle of reflection of 71.38 corresponding to a Bragg r reflection of order 4. Find the distance between atomic planes 66. Optics. Plastic, ur 5 20° for the salt crystal to the nearest hundredth of an angstrom. 67. Sprinkler. A water sprinkler is placed in the corner of the yard. If it disseminates water through an angle of 85°28r, For Exercises 63–66, refer to the following: how much of the corner is it missing? Have you ever noticed that if you put a stick in the water, it looks 68. Sprinkler. Sprinklers are set alongside a circle drive. If each bent? We know the stick didn’t bend. Instead, the light rays bent, sprinkler disseminates water through an angle of 36°19r, which made the image appear to bend. Light rays propagating what angle is covered by both sprinklers? from one medium (like air) to another medium (like water) expe- rience refraction, or “bending,” with respect to the surface. Light bends according to Snell’s law, which states:

ni sin1ui2 5 nr sin1ur2 36”19’ 36”19’

69. Obstacle Course. As part of an obstacle course, participants are required to ascend to the top of a ladder placed against a building and then use a rope to climb the rest of the way to the where roof. The distance traveled can be calculated using the formula

■ d 5 15 sin u 1 4 3, where u is the angle the ladder makes ni is the refractive index of the medium the light is leaving, the ! incident medium. with the ground and d is the distance traveled, measured in feet. Find the exact distance traveled by the participants if u 5 60°. ■ ui is the incident angle between the light ray and the normal (perpendicular) to the interface between mediums. 70. Obstacle Course. If the ladder in Exercise 69 is placed closer to the wall, the formula for distance traveled becomes ■ nr is the refractive index of the medium the light is entering. 15 ■ d 5 1 4. Approximate the distance traveled by the ur is the refractive angle between the light ray and the normal csc u (perpendicular) to the interface between mediums. participants if u 5 65°.

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71. Hot-Air Balloon. A hot-air balloon is tethered by ropes on of one tether by sin 60°, find the exact height of the balloon two sides that form a 45° angle with the ground. If the height when 100-foot ropes are used. of the balloon can be determined by multiplying the length 73. Staircase. The pitch of a staircase is given as 40°18r27s. of one tether by sin 45°, find the exact height of the balloon Write the pitch in decimal degrees. when 100-foot ropes are used. 74. Staircase. The height, measured in feet, of a certain staircase 72. Hot-Air Balloon. A hot-air balloon is tethered by ropes on is given by the formula h 5 15 tan u, where u is the pitch of two sides that form a 60° angle with the ground. If the height the staircase. What is the height of a staircase with a pitch of of the balloon can be determined by multiplying the length 39°28r37s?

• CATCH THE MISTAKE

In Exercises 75 and 76, explain the mistake that is made.

75. sec 60° 76. sec136°25r2

Solution: Solution:

Convert 36° 25r to 25 1 5 0.25 Write secant as the reciprocal sec u 5 decimal degrees. 100 of cosine. cos u

36° 25 5 36.25° 1 r Substitute u 5 60°. sec 60° 5 cos 60° Use the reciprocal 1 1 identity, sec u 5 . sec136.25°2 5 !3 1 cos u cos136.25°2 Recall cos 60° 5 . sec 60° 5 2 !3 Approximate with a 2 calculator. sec136.25°2 < 1.2400

2 This is incorrect. What mistake was made? Simplify. sec 60° 5 !3

2!3 Rationalize the denominator. sec 60° 5 3 This is incorrect. What mistake was made?

• CONCEPTUAL

In Exercises 77–80, determine whether each statement is true or false.

1 79. When approximating values of sin 10° and cos 10° with a calcu- 77. cos 30° 5 sec a b 30° lator, it is important for your calculator to be in degree mode. 80. tan120°50r2 5 cot170°10r2 78. sin 50° 5 0.77

For Exercises 81–84, refer to the following: Thus far in this text, we have only discussed trigonometric Use trigonometric ratios and the assumption that a is much values of acute angles, 0° , u , 90°. What about when u is larger than b to approximate the values without using a calculator. approximately 0° or 90°? We will formally consider these cases θ in the next chapter, but for now, draw and label a right triangle b that has one angle very close to 0° so that the opposite side is very a small compared to the adjacent side. Then the hypotenuse and the 81. sin 0° 82. cos 0° 83. cos 90° 84. sin 90° adjacent side are very close in size.

• CHALLENGE

sec 45° 1 tan 30° 1sin 45°21cot 30°2 85. Find the exact value of . 86. Find the exact value of . 2 cos30° csc 60°

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87. Find the exact value of cos 75° 2 1csc 45°21cos 30°2, tan125° 10r 15s2 1 sec146°14r 26s2 89. Calculate . csc123°17 23 2 !6 2 !2 r s given that sin 15° 5 . 4 cos133°38r1s2 90. Calculate 2 cot136°58r 6s2. 88. Find the exact value of cot 75° 1cos 45° 1 sin 30°2, given that csc150°33r2s2 tan 15° 5 2 2 !3.

• TECHNOLOGY

In Exercises 91 and 92, perform the indicated operations. Which gives you a more accurate value?

91. Calculate sec 70° the following two ways: 92. Calculate csc 40° the following two ways:

a. Write down cos 70° (round to three decimal places), a. Find sin 40° (round to three decimal places), and and then divide 1 by that number. Write the number to then divide 1 by that number. Write this last result to five decimal places. five decimal places.

b. First findc os 70° and then find its reciprocal. Round the b. First finds in 40° and then find its reciprocal. Round the result to five decimal places. result to five decimal places.

In Exercises 93 and 94, illustrate calculator procedures for converting between DMS and DD. 93. Convert 3°14r25s to decimal degrees. Round to three deci- 94. Convert 27.683° to degrees-minutes-seconds. Round to three mal places. decimal places.

1.5 SOLVING RIGHT TRIANGLES SKILLS OBJECTIVES CONCEPTUAL OBJECTIVES ■■ Identify the number of significant digits to express the ■■ Understand that the least accurate number in your lengths of sides and measures of angles when solving right calculation determines the accuracy of your result. triangles. ■■ Understand that when an acute angle is given, then the ■■ Solve right triangles given the measure of an acute angle third angle can be found exactly. The remaining unknown and the length of a side. side lengths can be found using right triangle trigonometry ■■ Solve right triangles given two side lengths. and the Pythagorean theorem. ■■ Understand that the trigonometric inverse keys on a calculator can be used to approximate the measure of an angle, given its trigonometric function value.

To solve a triangle means to find the measure of the three angles and three side lengths of 1.5.1 SKILL the triangle. In this section, we will discuss only right triangles. Therefore, we know one Identify the number of significant angle has a measure of 90°. We will know some information (the lengths of two sides or digits to express the lengths of the length of a side and the measure of an acute angle), and we will determine the measures sides and measures of angles of the unknown angles and the lengths of the unknown sides. However, before we start when solving right triangles. solving right triangles, we must first discuss accuracy and significant digits.

1.5.1 CONCEPTUAL 1.5.1 Accuracy and Significant Digits Understand that the least accurate number in your calculation If we are upgrading our flooring and quickly measure a room as 10 feet by 12 feet and determines the accuracy of we want to calculate the diagonal length of the room, we use the Pythagorean theorem. your result. WORDS MATH Apply the Pythagorean theorem. 102 1 122 5 d2 Simplify. d2 5 244 Use the square root property. d 5 6!244

d = ? ft 10 ft Be sure the length of the diagonal is positive. d 5 !244 Approximate the radical with 12 ft a calculator. d < 15.62049935

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Would you say that the 10-foot by 12-foot room has a diagonal of 15.62049935 feet? No, because the known room measurements were given only with an accuracy of a STUDY TIP foot and the diagonal above is calculated to eight decimal places. Your results are no The least accurate measure in your calculation determines the more accurate than the least accurate measure in your calculation. In this example we accuracy of your result. round to the nearest foot, and hence we say that the diagonal of the 10-foot by 12-foot room is about 16 feet. Significant digits are used to determine the precision of a measurement.

SIGNIFICANT DEFINITION Significant Digits NUMBER DIGITS The number of significant digits in a number is found by counting all of the digits 0.04 1 from left to right starting with the first nonzero digit. 0.276 3 0.2076 4 1.23 3 17 2 We have placed a question mark next to 8000 simply because we don’t know the 17.00 4 answer. If 8000 is a result from rounding to the nearest thousand, then it has one significant digit. If 8000 is the result of rounding to the nearest ten, then it has three 17.000 5 significant digits, and if there are exactly 8000 people surveyed, then 8000 is an exact 6.25 3 value and it has four significant digits. In this text, we will assume that integers have 8000 ? the greatest number of significant digits. Therefore, 8000 has four significant digits and can be expressed in scientific notation as 8.000 3 103. In solving right triangles, we first determine which of the given measurements has the least number of significant digits so we can round our final answers to the same number of significant digits.

[ CONCEPT CHECK] EXAMPLE 1 Identifying the Least Number of Significant Digits In solving a triangle, the least/ greatest number of significant Determine the number of significant digits corresponding digits of the given information to the given information in the following triangle: α is used when solving for the measure of an acute angle and a side length. In solving unknown side lengths and angle this right triangle, what number of significant digits measures. should be used to express the remaining angle measure c b ϭ 27.3 ft ▼ and side lengths? ANSWER Least

β ϭ 45º Solution: a Determine the significant digits corresponding tob 5 45°. Two significant digits Determine the significant digits corresponding tob 5 27.3 feet. Three significant digits In solving this triangle, the remaining side lengths of a and c and the measure of a should be expressed to two significant digits. 1.5.2 SKILL Solve right triangles given the measure of an acute angle and the length of a side. 1.5.2 Solving a Right Triangle Given an Acute Angle Measure and a Side Length 1.5.2 CONCEPTUAL Understand that when an acute When solving a right triangle, we already know that one angle has measure 90°. Let us angle is given, then the third now consider the case when the measure of an acute angle and a side length are given. angle can be found exactly. The Since the measure of one of the acute angles is given, the remaining acute angle can remaining unknown side lengths be found using the fact that the sum of three angles in a triangle is 180°. Right triangle can be found using right triangle trigonometry is then used to find the remaining side lengths. trigonometry and the Pythagorean theorem.

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EXAMPLE 2 Solving a Right Triangle Given an Angle and a Side Solve the right triangle—find a, b, and a. Solution:

STEP 1 Determine accuracy. α Since the given quantities (15 feet and 56°) both are expressed to two significant digits, 15 ft b we will round final calculated values to two significant digits. β = 56º a STEP 2 Solve for a. Two acute angles in a right triangle are complementary. a 1 56° 5 90° Solve for a. a 5 34°

STEP 3 Solve for a. The cosine of an angle is equal to the adjacent a STUDY TIP cos 56° 5 Make sure your calculator is in side over the hypotenuse. 15 “degrees” mode. Solve for a. a 5 15 cos 56° Evaluate the right side of the expression using a calculator. a < 8.38789

Round a to two significant digits. a < 8.4 ft

STEP 4 Solve for b. Notice that there are two ways to solve for b: trigonometric functions or the Pythagorean theorem. Although it is tempting to use the Pythagorean theorem, it is better to use the given information with trigonometric functions than to use a value that has already been rounded, which could make results less accurate.

The sine of an angle is equal to the b sin 56° 5 opposite side over the hypotenuse. 15

Solve for b. b 5 15 sin 56° Evaluate the right side of the expression using a calculator. b < 12.43556

Round b to two significant digits. b < 12 ft

STEP 5 Check the solution.

Angles and sides are rounded to two 34º significant digits.

15 ft 12 ft

56º 8.4 ft

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Check the trigonometric values of the specific angles by calculating the [ CONCEPT CHECK] trigonometric ratios. In Your Turn, the angle u can be ? 8.4 ? 12 ? 8.4 calculated exactly. Can the side sin 34° 5 cos 34° 5 tan 34° 5 15 15 12 lengths be calculated exactly? 0.5592 < 0.56 0.8290 < 0.80 0.6745 < 0.70 ▼ ANSWER No, they must be approximated. ▼

YOUR TURN Given the triangle shown here, solve the θ right triangle—find a, b, and u. ▼ 33 in. ANSWER b u 5 53°, a < 26 in., and b < 20 in. 37º a

1.5.3 Solving a Right Triangle Given the Lengths of Two Sides

When solving a right triangle, we already know that one angle has measure 908. Let us 1.5.3 SKILL now consider the case when the lengths of two sides are given. In this case, the third Solve right triangles given two side can be found using the Pythagorean theorem. If we can determine the measure of side lengths. one of the acute angles, then we can find the measure of the third acute angle using the fact that the sum of the three angle measures in a triangle is 1808. How do we find the 1.5.3 measure of one of the acute angles? Since we know the side lengths, we can use right CONCEPTUAL triangle ratios to determine the trigonometric function (sine, cosine, or tangent) values Understand that the trigonometric and then ask ourselves: What angle corresponds to that value? inverse keys on a calculator Sometimes, we may know the answer exactly. For example, if we determine that can be used to approximate the 1 1 measure of an angle, given its sin u 5 2, then we know that the acute angle u is 308 because sin 30° 5 2. Other times we trigonometric function value. may not know the corresponding angle, such as sin u 5 0.9511. Calculators have three keys (sin21, cos21, and tan21) that help us determine the unknown angle. For example, a

calculator can be used to assist us in finding what angle u corresponds to sin u 5 0.9511. sin2110.95112 5 72.00806419 At first glance, these three keys might appear to yield the reciprocal; however, the 21 superscript corresponds to an inverse function. We will learn more about inverse trigonometric functions in Chapter 6, but for now we will use these three calculator keys to help us solve right triangles.

EXAMPLE 3 Using a Calculator to Determine [ CONCEPT CHECK] an Acute Angle Measure When two side lengths are given Use a calculator to findu . Round answers to the nearest degree. but the two acute angles in a right triangle are unknown, how do we a. cos u 5 0.8734 find an approximation of the acute b. tan u 5 2.752 angles? Solution (a): ▼ ANSWER Inverse trigonometric function Use a calculator to evaluate the inverse cosine function. u 5 cos2110.87342 5 29.14382196° Round to the nearest degree. u < 29° Solution (b): Use a calculator to evaluate the inverse tangent function. u 5 tan2112.7522 5 70.03026784° Round to the nearest degree. u < 70°

▼ ▼ YOUR TURN Use a calculator to findu , given sin u 5 0.7739. Round the answer ANSWER to the nearest degree. 518

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EXAMPLE 4 Solving a Right Triangle Given Two Sides Solve the right triangle—find a, a, and b. Solution: α STEP 1 Determine accuracy. 37.21 cm The given sides have four significant 19.67 cm digits; therefore, round final calculated values to four significant digits. β a

STEP 2 Solve for a. The cosine of an angle is equal to the 19.67 cm cos a 5 adjacent side over the hypotenuse. 37.21 cm

Evaluate the right side with a calculator. cos a < 0.528621338 Write the angle a in terms of the inverse cosine function. a < cos2110.5286213382 Use a calculator to evaluate the inverse cosine function. a < 58.08764855° Round a to four significant digits. a < 58.09°

STEP 3 Solve for b. The two acute angles in a right triangle STUDY TIP are complementary. a 1 b 5 90° To find a length of a side in a Substitute a < 58.09°. 58.09 1 b < 90° right triangle, use sine, cosine, and tangent functions when an Solve for b. b < 31.91° angle measure is given. The answer is already rounded to four significant digits. To find the measure of an angle in a right triangle, given the STEP 4 Solve for a. proper ratio, use inverse sine, Use the Pythagorean theorem since inverse cosine, and inverse 2 2 2 tangent functions when the the lengths of two sides are given. a 1 b 5 c lengths of the sides are known. Substitute given values for b and c. a2 1 19.672 5 37.212 Solve for a. a < 31.5859969 Round a to four significant digits. a < 31.59 cm

STEP 5 Check the solution. 58.09º 37.21 cm Angles are rounded to the nearest hundredth 19.67 cm degree, and sides are rounded to four significant digits of accuracy. 31.91º Check the trigonometric values of the specific 31.59 cm angles by calculating the trigonometric ratios.

? 19.67 ? 31.59 sin131.91°2 5 sin158.09°2 5 37.21 37.21 0.5286 < 0.5286 0.8489 < 0.8490

▼ ▼ ANSWER YOUR TURN Solve the right triangle—find a, a, and b. α a < 16.0 mi, a < 43.0°, and b < 47.0° 23.5 mi 17.2 mi

β a

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Applications In many applications of solving right triangles, you are given the length of a side and the measure of an acute angle and asked to find one of the other sides. Two common examples involve an observer (or point of reference) located on the horizontal and an object that is either above or below the horizontal. If the object is above the horizontal, then the angle made is called the angle of elevation, and if the object is below the horizontal, then the angle made is called the angle of depression. For example, if a race car driver is looking straight ahead (in a horizontal line of sight), then looking up is elevation and looking down is depression.

Angle of elevatio

Horizontal

Angle of depression Calvin Chan/Shutterstock56817232

If the angle is a physical one (like a skateboard ramp), then the appropriate name is the angle of inclination.

Angle of inclination

EXAMPLE 5 Angle of Depression (NASCAR) In this picture, car 19 is behind the leader car 2. If the angle of depression is 18° from the car 19’s driver’s eyes to the bottom of the 3-foot high back end of car 2 (side opposite the angle of depression), how far apart are their bumpers? Assume that the horizontal distance from the car 19’s driver’s eyes to the front of his car is 5 feet.

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Solution: x 18º Draw an appropriate right triangle and 3 ft label the known quantities.

Because the sides of interest are adjacent 3 ft and opposite to the known angle, tan 18° 5 identify the tangent ratio. x 3 ft Solve for x. x 5 tan18° Evaluate the right side. x < 9.233 ft Round to the nearest foot. x < 9 ft Subtract 5 feet from x. 9 2 5 5 4 ft

Their bumpers are approximately 4 feet apart .

Suppose NASA wants to talk with astronauts on the International Space Station (ISS), which is traveling at a speed of 17,700 mph, 400 kilometers (250 miles) above the surface of the Earth. If the antennas at the ground station in Houston have a pointing 1 ° error of even 1 minute, that is 1r 5 a b 5 0.01667°, they will miss the chance to talk with the astronauts. 60

EXAMPLE 6 Pointing Error Assume that the ISS (which is 108 meters long and 73 meters wide) is in a 400-kilometer low Earth orbit. If the communications antennas have a 1-minute pointing error, how many meters “off” will the communications link be? Solution: Draw a right triangle that depicts this scenario.

x 1´ 400 km ISS

(NOT TO SCALE)

Because the sides of interest are adjacent x and opposite to the known angle, tan11r2 5 identify the tangent ratio. 400 km

Solve for x. x 5 1400 km2 tan11r2 1 ° Convert 1r to decimal degrees. 1r 5 a b < 0.016666667° 60 Convert the equation for x in terms of decimal degrees. x < 1400 km2 tan10.016666667°2 Evaluate the expression on the right. x < 0.116355 km < 116 m 400 kilometers is accurate to three significant digits. So we express the answer to three significant digits. The pointing error causes the signal to be off by approximately 116 meters . Since the ISS is only 108 meters long, it is expected that the signal will be missed by the astronaut crew.

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In navigation, the word bearing means the direction to which a vessel is pointed. Heading is the direction in which the vessel is actually traveling. Heading and bearing are only synonyms when there is no wind on land. Direction is often given as a bearing, which is the measure of an acute angle with respect to the north-south vertical line. “The

plane has a bearing N 20° E” means that the plane is pointed 20° to the east of due north.

N 20º E

20º W E

EXAMPLE 7 Bearing (Navigation)

A jet takes off bearing N 28° E and flies 5 miles, and then makes a left 190°2 turn and flies 12 miles farther. If the control tower operator wanted to locate the plane, what bearing would she use? Round to the nearest degree. Solution: Draw a picture that represents this scenario. N

12 mi

5 mi

28º

12 Identify the tangent ratio. tan u 5 5

21 12 Use the inverse tangent function to solve for u. u 5 tan a b < 67.4° 5 Subtract 28° from u to find the bearing,b . b < 67.4° 2 28° < 39.4°

Round to the nearest degree. b < N 39° W

[SECTION 1.5] SUMMARY

In this section, we have solved right triangles. When either a side lengths and angle measures). The least accurate number used in length and an acute angle measure are given or two side lengths are your calculations determines the appropriate number of significant given, it is possible to solve the right triangle (find all unknown side digits for your results.

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[SECTION 1.5] EXERCISES

• SKILLS

In Exercises 1–4, determine the number of significant digits corresponding to each of the given angle measures and side lengths. 1. a 5 37.5° 2. b 5 49.76° 3. a 5 0.37 km 4. b 5 0.2 mi

In Exercises 5–10, use a calculator to find the measure of angleu . Round answer to the nearest degree.

5. sin u 5 0.7264 6. sin u 5 0.1798 7. cos u 5 0.5674 8. cos u 5 0.8866 9. tan u 5 8.235 10. tan u 5 3.563

In Exercises 11–30, refer to the right triangle diagram and the given information to find the indicated measure. Write your answers for angle measures in decimal degrees. 11. b 5 35°, c 5 17 in.; find a. 12. b 5 35°, c 5 17 in.; find b. α 13. a 5 55°, c 5 22 ft; find a. 14. a 5 55°, c 5 22 ft; find b. 15. a 5 20.5°, b 5 14.7 mi; find a. 16. b 5 69.3°, a 5 0.752 mi; find b. c 17. b 5 25°, a 5 11 km; find c. 18. b 5 75°, b 5 26 km; find c. b 19. a 5 48.25°, a 5 15.37 cm; find c. 20. a 5 29.80°, b 5 16.79 cm; find c. 21. a 5 29 mm, c 5 38 mm; find a. 22. a 5 89 mm, c 5 99 mm; find b. β a 23. b 5 2.3 m, c 5 4.9 m; find a. 24. b 5 7.8 m, c 5 13 m; find b

25. a 5 21° 17r, b 5 210.8 yd; find a. 26. b 5 27° 21r, a 5 117.0 yd; find b.

27. b 5 15° 20r, a 5 10.2 km; find c. 28. b 5 65° 30r, b 5 18.6 km; find c.

29. a 5 40° 28r10s, a 5 12,522 km; find c. 30. a 5 28° 32r50s, b 5 17,986 km; find c.

In Exercises 31–48, refer to the right triangle diagram and the given information to solve the right triangle. Write your answers for angle measures in decimal degrees. 31. a 5 32° and c 5 12 ft 32. a 5 65° and c 5 37 ft α 33. a 5 44° and b 5 2.6 cm 34. a 5 12.0° and b 5 10.0 m

35. a 5 60° and c 5 5 in. 36. a 5 9.67° and c 5 5.38 in. c 37. b 5 72° and c 5 9.7 mm 38. b 5 45° and c 5 7.8 mm b 39. a 5 54.2° and a 5 111 mi 40. b 5 47.2° and a 5 9.75 mi 41. b 5 458, b 5 10.2 km 42. b 5 85.5°, b 5 14.3 ft β a 43. a 5 28°23r and b 5 1734 ft 44. a 5 72°59r and a 5 2175 ft 45. a 5 42.5 ft and b 5 28.7 ft 46. a 5 19.8 ft and c 5 48.7 ft 47. a 5 35,236 km and c 5 42,766 km 48. b 5 0.1245 mm and c 5 0.8763 mm

• APPLICATIONS

5 ft

(NOT TO SCALE) 1º

49. Golf. If the flagpole that a golfer aims at on a green measures Exercises 51 and 52 illustrate a midair refueling scenario that 5 feet from the ground to the top of the flag and a golfer military aircraft often enact. Assume the elevation angle that measures a 1° angle from top to bottom of the pole, how far the hose makes with the plane being fueled is u 5 36°. (in horizontal distance) is the golfer from the flag? Round to the nearest foot. 50. Golf. If the flagpole that a golfer aims at on a green measures a 5 feet from the ground to the top of the flag and a golfer Hose measures a 3° angle from top to bottom of the pole, how far θ = 36º (in horizontal distance) is the golfer from the flag? Round to b the nearest foot. 51. Midair Refueling. If the hose is 150 feet long, what should be the altitude difference a between the two planes? Round to the nearest foot.

Young_Trigonometry_6161_ch01_pp036-063.indd 52 15/12/16 3:04 PM 1.5 Solving Right Triangles 53

52. Midair Refueling. If the smallest acceptable altitude difference For Exercises 59–62, refer to the following: a between the two planes is 100 feet, how long should the hose Geostationary orbits are useful because they cause be? Round to the nearest foot. a satellite to appear stationary with respect to a Exercises 53–56 are based on the idea of a glide slope (the fixed point on the rotating Earth. As a result, an angle the flight path makes with the ground). antenna (dish TV) can point in a fixed direction and maintain a link with the satellite. The satellite Precision Approach Path Indicator (PAPI) lights are used as a visual orbits in the direction of the Earth’s rotation at an approach slope aid for pilots landing aircraft. The typical glide path altitude of approximately 35,000 kilometers. for commercial jet airliners is 38. A Navy fighter jet has an outer 59. Dish TV. If your dish TV antenna has a glide approach of 188–208. PAPI lights are typically configured pointing error of 1s (1 second), how long 35,000 km as a row of four lights. All four lights are on, but in different would the satellite have to be to maintain a combinations of red or white. If all four lights are white, then the link? Round your answer to the nearest meter. angle of descent is too high; if all four lights are red, then the angle of descent is too low; and if there are two white and two red, then 60. Dish TV. If your dish TV antenna has a 1s the approach is perfect. pointing error of 2 (half a second), how long would the satellite have to be to PAPI maintain a link? Round your answer to the 3º nearest meter. Runway Ground Altitude 61. Dish TV. If the satellite in a geostationary orbit (at 35,000 kilometers) was only 53. Glide Path of a Commercial Jet Airliner. If a commercial 10 meters long, about how accurately jetliner is 5000 feet (about 1 mile ground distance) from the pointed would the dish have to be? Give the runway, what should be the altitude of the plane to achieve answer in degrees to two significant digits. 2 red/2 white PAPI lights? (Assume that this corresponds to a 62. Dish TV. If the satellite in a geostationary 3° glide path.) orbit (at 35,000 kilometers) was only 30 meters long, about 54. Glide Path of a Commercial Jet Airliner. If a commercial how accurately pointed would the dish have to be? Give the jetliner is at an altitude of 450 feet when it is 5200 feet from answer in degrees to two significant digits. the runway (approximately 1 mile ground distance), what 63. Angle of Elevation (Traffic). A person driving in a sedan is is the glide slope angle? Will the pilot see white lights, red driving too close to the back of an 18 wheeler on an interstate lights, or both? highway. He decides to back off until he can see the entire 55. Glide Path of a Navy Fighter Jet. If the pilot of a Navy truck (to the top). If the height of the trailer is 15 feet and the fighter jet is at an altitude of 3000 feet when she is 15,500 sedan driver’s angle of elevation (to the top of the trailer from feet (approximately 3 miles ground distance) from the landing the horizontal line with the bottom of the trailer) is roughly facility, what is her glide slope angle (round to the nearest 30°, how far is he sitting from the end of the trailer? degree)? Is she too high or too low? 56. Glide Path of a Navy Fighter Jet. If the same pilot in 15 ft Exercise 55 raises the nose of a Navy fighter jet so that she 30Њ drops only 500 feet by the time she is 7800 feet from the x landing strip (ground distance), what is her glide angle at that time (round to the nearest degree)? Is she within the specs 64. Angle of Depression (Opera). The balcony seats at the opera 118°220°2 to land the jet? house have an angle of depression of 55° to center stage. If the horizontal (ground) distance to the center of the stage is In Exercises 57 and 58, refer to the illustration below that 50 feet, how far are the patrons in the balcony from the singer shows a search and rescue helicopter with a 308 field of view at center stage? with a search light.

50 ft 30º 55º

57. Search and Rescue. If the search and rescue helicopter is flying at an altitude of 150 feet above sea level, what is

the diameter of the circle illuminated on the surface of the Foto World/The Image Bank/Getty Images, Inc. water? 65. Angle of Inclination (Skiing). The angle of inclination of a 58. Search and Rescue. If the search and rescue helicopter is mountain with triple black diamond ski paths is 65°. If a skier flying at an altitude of 500 feet above sea level, what is the at the top of the mountain is at an elevation of 4000 feet, how diameter of the circle illuminated on the surface of the water? long is the ski run from the top to the base of the mountain?

Young_Trigonometry_6161_ch01_pp036-063.indd 53 15/12/16 3:04 PM 54 CHAPTER 1 Right Triangle Trigonometry

21 ft 18 ft 21 ft 21 ft 18 ft 21 ft Side line Post 4 ft 6 in. Alley line Right service court Left service court 13 ft 6 in.

Center Net service line Fore court Service line Back court Back screen

36 ft 27 ft Center mark Doubles Singles 13 ft 6 in. Right service court Left service court Baseline

42 ft

78 ft Side screen

66. Bearing (Navigation). If a plane takes off bearing N 33° W right service court (or top right corner of the left service court). and flies 6 miles and then makes a right turn 190°2 and flies When attempting this serve, a player will toss the ball rather high 10 miles farther, what bearing will the traffic controller use to into the air, bring the racquet back, and then make contact with the locate the plane? ball at the precise moment when the position of the ball in the air 67. Bearing (Navigation). If a plane takes off bearing N358E coincides with the top of the netted part of the racquet when the and flies 3 miles and then makes a left turn1 9082 and flies player’s arm is fully stretched over his or her head. 8 miles farther, what bearing will the traffic controller use to 69. Tennis. Assume that the player is serving into the right locate the plane? service court and stands just 2 inches to the right of the center 68. Bearing (Navigation). If a plane takes off bearing N488W line behind the baseline. If, at the moment the racquet strikes and flies 6 miles and then makes a right turn1 9082 and flies the ball, both are 72 inches from the ground and the serve 17 miles farther, what bearing will the traffic controller use to actually hits the top left corner of the right service court, locate the plane? determine the angle at which the ball meets the ground in the right service court. Round to the nearest degree. For Exercises 69 and 70, refer to the following: 70. Tennis. Assume that the player is serving into the right With the advent of new technology, tennis racquets can now be service court and stands just 2 inches to the right of the center constructed to permit a player to serve at speeds in excess of line behind the baseline. If the ball hits the top left corner of 120 mph. One of the most effective serves in tennis is a power the right service court at an angle of 448, at what height above serve that is hit at top speed directly at the top left corner of the the ground must the ball be struck?

For Exercises 71 and 72, refer to the following: 72. Chemistry. Now, suppose one of the chlorides (Cl) is removed. The resulting structure is triagonal in nature, resulting in the The structure of molecules is critical to the study of materials figure below. Does the angle u change? If so, what is its new science and organic chemistry, and has countless applications to value? a variety of interesting phenomena. Trigonometry plays a critical role in determining bonding angles of molecules. For instance, the 23 structure of the 1FeCl4Br22 ion (dibromatetetrachlorideferrate III) is shown in the figure below. θ Br 71. Chemistry. Determine the angle u [i.e., the angle between the axis containing the apical bromide atom (Br) and the segment connecting Br to Cl]. 2.354

2.097 θ Fe Br Cl Cl 120º 120º 2.354 Cl Cl Cl

2.219 Br Fe Cl Cl 73. Navigation. A plane takes off headed due north. With the

wind, the airplane actually travels on a heading of N 15° E. After traveling for 100 miles, how far north is the plane from its starting position?

Young_Trigonometry_6161_ch01_pp036-063.indd 54 15/12/16 3:04 PM 1.5 Solving Right Triangles 55

74. Navigation. A boat must cross a 150-foot river. While the shape of the canal to the nearest degree. Has erosion affected boat is pointed due east, perpendicular to the river, the current the shape of the canal? Explain. causes it to land 25 feet down river. What is the heading of the boat? For Exercises 77 and 78, refer to the following: After breaking a femur, a patient is placed in traction. The end of For Exercises 75 and 76, refer to the following: a femur of length l is lifted to an elevation forming an angle u with A canal constructed by a water-users association can be the horizontal (angle of elevation). approximated by an isosceles triangle (see the figure below). When the canal was originally constructed, the depth of the canal was 5.0 feet and the angle defining the shape of the canal was 608. Length Width Elevation θ Depth θ

77. Health/Medicine. A femur 18 inches long is placed into 75. Environmental Science. If the width of the water surface traction, forming an angle of 158 with the horizontal. Find today is 4.0 feet, find the depth of the water running through the height of elevation at the end of the femur. the canal. 78. Health/Medicine. A femur 18 inches long is placed in 76. Environmental Science. One year later, a survey is traction with an elevation of 6.2 inches. What is the angle of performed to measure the effects of erosion on the canal. It elevation of the femur? is determined that when the water depth is 4.0 feet, the width of the water surface is 5.0 feet. Find the angle u defining the

• CATCH THE MISTAKE For Exercises 79 and 80, refer to the right triangle diagram below and explain the mistake that is made. 80. If b 5 56° and c 5 15 feet, find b and then find a. α

c Solution: b Write sine as the opposite side b over the hypotenuse. sin 56° 5 β 15 a Solve for b. b 5 15 sin 56° 79. If b 5 800 feet and a 5 10 feet, findb . Use a calculator to approximate b. b < 12.4356 Solution: Round the answer to two significant digits. b < 12 ft Represent tangent as the opposite side over the b Use the Pythagorean theorem tan b 5 adjacent side. a to find a. a2 1 b2 5 c2

Substitute b 5 12 feet Substitute b 5 800 feet 800 2 2 2 tan b 5 5 80 and c 5 15 feet. a 1 12 5 15 and a 5 10 feet. 10 Solve for a. a 5 9

Use a calculator to evaluate b. b 5 tan 80° < 5.67° Round the answer to two 5 This is incorrect. What mistake was made? significant digits. a 9.0 ft Compare this with the results from Example 1. Why did we get a different value for a here?

• CONCEPTUAL In Exercises 81–88, determine whether each statement is true or false. 81. If you are given the lengths of two sides of a right triangle, 83. If you are given the measures of the two acute angles of a you can solve the right triangle. right triangle, you can solve the right triangle. 82. If you are given the length of one side and the measure of 84. If you are given the length of the hypotenuse of a right triangle one acute angle of a right triangle, you can solve the right and the measures of the angle opposite the hypotenuse, you triangle. can solve the right triangle.

Young_Trigonometry_6161_ch01_pp036-063.indd 55 15/12/16 3:04 PM 56 CHAPTER 1 Right Triangle Trigonometry

85. The number 0.0123 has five significant digits. 88. If you are given the lengths of two sides of a right triangle, 86. If the measurement 700 feet has been rounded to the nearest you will need to use an inverse trigonometric function to find whole number, it has three significant digits. the third side length. 87. If you are given the length of one side and the measure of one acute angle, you can solve the right triangle using either the sine or cosine function.

• CHALLENGE 89. Use the information in the picture below to determine the 91. From the top of a 12-foot ladder, the angle of depression to the height of the mountain. far side of a sidewalk is 45°, while the angle of depression to the near side of the sidewalk is 65°. How wide is the sidewalk? 92. Find the perimeter of triangle A.

x 10

5 51º38º 48º A y 1900 ft

93. Solve for x. 90. Two friends who are engineers at Kennedy Space Center (KSC) watched the shuttle launch. Carolyn was at the Vehicle Assembly Building (VAB) 3 miles from the launch pad and x Jackie was across the Banana River, which is 8 miles from the launch pad. They called each other at liftoff, and after 10 seconds they each estimated the angle of elevation with 12º respect to the ground. Carolyn thought the angle of elevation was approximately 40° and Jackie thought the angle of elevation was approximately 15°. Approximately how high 35º was the shuttle after 10 seconds? (Average their estimates 5.33 and round to the nearest mile.) 94. An electric line is strung from a 20-foot pole to a point Carolyn 12 foot up on the side of a house. If the pole is 250 feet from the house, what angle does the electric line make with the pole? Jackie

15º 40º 5 mi 3 mi Image Bank/Getty Images, Inc.; Photonica/Getty Images

• TECHNOLOGY 21 21 95. Use a calculator to finds in 1sin 40°2. 99. Based on the result from Exercise 95, what would sin 1sin u2 21 be for an acute angle u? 96. Use a calculator to findc os 1cos 17°2. 21 21 100. Based on the result from Exercise 96, what would cos 1cos u2 97. Use a calculator to findc os1cos 0.82. be for an acute angle u? 21 98. Use a calculator to finds in1sin 0.32.

Young_Trigonometry_6161_ch01_pp036-063.indd 56 15/12/16 3:04 PM CHAPTER 1 REVIEW 57 15/12/16 3:04 PM

Chapter Review Chapter < 180º α θ = 180º

θ β Obtuse angle 60º 90º <

x + 2

α x Initial side β Negative angle = 90º θ Supplementary angles 30º-60º-90º 30º side erminal x T Right angle: Quarter rotation 3 √ 2 c = 180º =

2 γ b + + β 2 a + α 45º x 2 √ x β c = 90º < 90º b θ β 45º-45º-90º + α Positive angle Initial side α θ α 0º < Acute angle 45º γ β side erminal Complementary angles x a One complete rotation = 360º T KEY IDEAS/FORMULAS Special right triangles Triangles Angles, degrees, and triangles Angles, degrees, measure Angles and degree CONCEPT ] 1 REVIEW CHAPTER 1.1 [ SECTION Young_Trigonometry_6161_ch01_pp036-063.indd 57 Young_Trigonometry_6161_ch01_pp036-063.indd 58 CHAPTER 1 REVIEW 1.3 1.2 SECTION 58 CHAPTER 1 CONCEPT Right triangleratios Trigonometric functions: functions: Righttriangleratios Definition 1oftrigonometric Classification oftriangles Finding anglemeasuresusinggeometry Similar triangles Cofunctions Right Triangle Trigonometry KEY IDEAS/FORMULAS SOH in arighttriangle. Definition 1definestrigonometricfunctionsofacuteanglesasratiosthelengthsides Congruent triangles:Sameshapeandsize Similar triangles:Sameshape Reciprocal identities: TOA CAH If a a r a 5 1 c m b o b b t

u ba n 5 5 c 5 t s a o i c n n 7 9 c s t r

0 u u u a ° 1 n 5 5 , then: 5 8

u 6 c

h h o a 3 d y y p o a c 1 j p p p d p s a o o o c j p s c t s a 4

t t s a u o e e i e e c 2 i n n c s n t n n e 5 e i

a a a t n u u t e t s s 5 5 5 s e e m n i 1 n c c c

u o s o

c s t

b b b s e c

u 5 b' c Hypotenuse o 1 s

c' u Adjacent θ c a a' b Opposite 15/12/16 3:04 PM CHAPTER 1 REVIEW 59 15/12/16 3:04 PM

. ° 0 8 1 Chapter Review Chapter 5 g 1 b 1 a u r 3 2 2 2 1 2 1 ! " 5 COS s and reciprocal identities. 0 u u

6 s n i o ° s c 1 or ° u 3 2 B 5 2 1 2 2 5 0 , to get csc, sec, and cot, respectively. ! 0 r " 1 u

6 0 21 3 n SIN A 6 a t 5 or r ° B B 0 0 1 1 6 6 A A 5 u 308 458 608 5 The third angle measure can be found exactly using The third angle measure can be found exactly using of the Right triangle trigonometry is used to find the lengths remaining sides. Pythagorean theorem. The length of the third side can be found using the right The measures of the acute angles can be found using triangle trigonometry. r s

1 1 n n n n The other trigonometric function values can be found for these The other trigonometric angles using DMS to DD: Divide by multiples of 60. DD to DMS: Multiply by multiples of 60. results. Always use given measurements if possible for accurate of answer. Pay attention to proper significant digits for accuracy Make sure the calculator is in degree mode. Make sure the calculator the reciprocal The sin, cos, and tan buttons can be combined with button, 1/x or x KEY IDEAS/FORMULAS ° 0 6

d n a

°, 5 4

°, 0 3 Representing partial degrees: DD or DMS Solving right triangles Accuracy and significant digits Solving a right triangle given an acute angle measure and a side length Evaluating trigonometric functions: trigonometric functions: Evaluating with calculators Exactly and functions exactly Evaluating trigonometric measures: for special angle Using calculators to evaluate (approximate) Using calculators to evaluate trigonometric function values Solving a right triangle given the lengths of two sides CONCEPT 1.5 SECTION 1.4 Young_Trigonometry_6161_ch01_pp036-063.indd 59 Young_Trigonometry_6161_ch01_pp036-063.indd 60 REVIEW EXERCISES [ indicated angle. Find themeasureof angles. Find (a)thecomplementand(b)supplementofgiven Refer tothefollowing308-60-90triangle. 21. 20. 19. 1.2 18. 17. Applications 16. 15. 14. 13. 12. 11. 10. 9. 8. 7. 6. 5. Refer tothefollowingtriangle. 1. 1.1 60 Refer tothefollowing458-45-90triangle. Refer tothefollowingrighttriangle. CHAPTER 1 2 Similar Triangles Angles, Degrees,andTriangles find find / / / second handsweepsinexactly 15seconds? Clock. What isthemeasure(indegrees) oftheanglethat minute handsweepsinexactly 25minutes? Clock. What isthemeasure(indegrees) oftheanglethat of thetwo legs? kilometers, whatarethelengths If thehypotenuse haslength12 and thehypotenuse? what arethelengthsofotherleg If theshorterleg haslength3feet, 2 If thehypotenuse haslength the hypotenuse? 12 yards,how longis If thetwo legs have length If If If If If If If If all threeangles. all threeangles. 8 ! CHAPTER 1 ° F F F a a b a g g a a

2 5 5 5 5 5 5 5 5 5 g g 5 5

2. f . . e

1 7 9 4 7 7 7 b b 1 1 e 1 and and and 0 5 5 5 t 0 2 and and , 7 and ° ° ° how longarethelegs? 5 0 , , , ° ° ° find find find

and and b c c a a 3. b Right Triangle Trigonometry 5 5 5 5 5

/ / / 5 b b 3 1 1 4 5 C. D G 6 7 5 2 5 5 8 , ° findc. b b . . , , ,

finda. findb. findc. , , 2 3 find find REVIEW EXERCISES] 4. 5 5 ° °

, ,

7 8 ° m γ || n a x √ 3x 45º GH EF α 30 α + º β CD + AB γ x b x √ =180º 2x c 2 60 x β º 45º m n encounter inyouranswers,butleaveanswersexact. function values.Rationalizeanydenominatorsthatyou Use thefollowingtriangletofindindicatedtrigonometric 30. 29. Applications 28. 27. 26. 25. are similar. Calculate thespecifiedlengthsgiventhattwotriangles 24. 23. 22. 38. 37. 36. 35. 34. 33. 1.3 32. 31. have anindicateddistanceof3feet,measuring1inchwitharuler. In ahomeremodelingproject,yourarchitectgivesyouplansthat For Exercises31and32,refertothefollowing: Triangle Ratios Definition 1ofTrigonometricFunctions:Right the NBA center? shadow, andhis4-footsoncastsa1-footshadow, how tallis Height ofaMan.IfanNBA centercastsa1foot9inch backboard measures1.2meters.How tallisthetree? At thesametimeofday, theshadow ofa4-meterbasketball Height ofa Tree. The shadow ofatreemeasures9.6meters. F E C F E / / / c t c s s c 1 Home Renovation. How wideisthepantryifitmeasures measures Home Renovation. How wideisthebuilt-in refrigeratorifit D D A A A a e i o s o 4 1 n n F F F c c incheswitharuler? 5 5 5 5 t s 5 5 5 5 5 5

u u

u u u 5 5 5 1 8 8 4 8 1 1 ? 4 5

4 . , 1 5 0 . , c 2

7 7 7

, , E m c

5 5 5

m B B m m m ° ° ° , 1 5 5

, , , , , , 3 1 5 5

find find find ,

incheswitharuler? C ? ? 1 8 5 2 ,

/ / / ,

8 A B E

m . . . ,

B 5 ? A θ 2 B Refrigerator C

3 Pantry D E F

15/12/16 3:04 PM REVIEW EXERCISES b b r 61 α α n i 15/12/16 3:05 PM n

º r θ º i a θ a 1.00. ve c c 5 Angle i Incident n Angle β β Refracti Review Exercises Review Ray Light Surface 2 a r u 30°. 1

of the indicated refractive of 5 60°. km sin r

r n u b n 5

ace find c. Hose

ft 5 u find c.

θ = 30Њ ft 2

i 32,525 find a. mi yd, cm,

u 36.09°

1 km,

5 mm 2154 mi,

ft 45.7 5

find a.

35.26° entering. c 215 sin

find a. r i 5 mediums. u 8.5 21 5 120.0 96.5 n 19.22 5 in., ft, 5

b 21.9 r b and 5 a 5 5 5 u 5 15 27 5 c c a b a

km , , and

5 b 5 r r r and

and index of refraction index of c c ft

and and 25°, 37°45 75°10 65° 50°, 33.5°, 47.45°, 30° 48.5° 30°15 30.5 11,798 Air has an index of refraction value of of refraction value Air has an index The incidence angle is 5 5 5 5 5 5 5 5 5 5 is the refractive angle is the refractive is the refractive index of the index is the refractive is the incident angle between is the incident angle 5 5 is the refractive index of the is the refractive i r i r

medium the light is leaving. medium the medium the light is the light ray and the normal the light ray and the interf (perpendicular) to the between mediums. between the light ray and the between the light ray to the normal (perpendicular) between interface n u n u The incident medium is air. ■ ■ find b. Optics. Glass, b a a a b b a b a a a a Optics. Glycerin, Solving Right Triangles

■ ■ ■ ■ ■ medium given the following assumptions: Use the right triangle diagram below and the information given to solve the right triangle. Write your answers for angle measures in decimal degrees. Applications law, which states according to Snell’s Light bends 85. 1.5 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. Applications The illustration below shows a midair refueling scenario that our military aircraft often use. Assume the elevation angle that the hose makes with the plane being fueled is 86. Calculate the Use the right triangle diagram below and the information Use the right triangle diagram below and Write your answers for given to find the indicated measure. angle measures in decimal degrees. 2 2 17.3° 16.8° 60° 45° 1 60° 60°

1 60° 60°

2 2 sec cot cos sec csc cos cos tan A u

s 2 1 2 2 58. 61. 52. 64. 67. 70. 49. 55. 15

! 8359) r r 8179)

55° 60° c. 15 45

1 1 2

cos(42 sin(20 68° 25° cos sec 1 2

b. 82. 84. 74. 76. 44. 46. 19.76° , to calculate the following values. 57° 30° 45° 45°

33° 45° 45°

1 45°

u u

3 cos cot cot cot cos sin tan csc sec 2 sin cos

63. 54. 66. 48. 51. 57. 60. 69. 72. a. u 5

round to the nearest minute round to the nearest minute round to the nearest second round to the nearest second

2 2

x 2 sec t cot cos

s 2 sin 2 2

5 5 5 25

r r 5 40.25° 45° 25.2° 30° 17 30 A

30° 30° 60°

1 1 30° 30° 45°

1 42° 30° 60° 30°

cos(618489) 30.175° 25.258° sin(378159) 39° 29° 42.25° 60.45° csc sin tan csc sec cot tan sin sin csc cos tan csc sin sin Evaluating Trigonometric Functions: Exactly and Evaluating Trigonometric with Calculators

83. 79. 80. 81. 73. 75. Convert from decimal degrees to degrees-minutes-seconds. 77. 78. 71. to decimal degrees. Convert from degrees-minutes-seconds Round to the nearest hundredth if only minutes are given and to the nearest thousandth if seconds are given. Use a calculator to approximate the following trigonometric Use a calculator to approximate the following decimal places. function values. Round answers to four 65. 68. 56. 59. 62. tient identity, 53. reciprocal identitiesUse the results in Exercises 47–55 and the to calculate the following values. 47. 50. the trigonometric quo Use the results in Exercises 47–52 and 45. 1.4 40. 41. 42. function in terms of its cofunction. Write the trigonometric 43. Use the cofunction identities to fill in the blanks. to fill identities cofunction Use the 39. Use a calculator to approximate the following trigonometric function values. Round answers to four decimal places. Label each trigonometric function value with the correspondingLabel each trigonometric value (a)–(c). Young_Trigonometry_6161_ch01_pp036-063.indd 61 Young_Trigonometry_6161_ch01_pp036-063.indd 62 REVIEW EXERCISES 104. 103. decimal places. Assume a308-60-90triangle.Roundyouranswerstotwo Section 1.1 Technology Exercises 102. 101. 100. 62 99.

lengths oftheotherleg andthehypotenuse? If thelongerleg haslength87.65centimeters,whatarethe of theotherleg andhypotenuse? If theshorterleg haslength41.32feet,whatarethelengths to locatetheplane? 9 milesfarther, whatbearingwillthetraffic controlleruse and fl Bearing (Navigation). Ifaplanetakes off bearingN to locatetheplane? 10 milesfarther, whatbearingwillthetraffic controlleruse and fl Bearing (Navigation). Ifaplanetakes off bearingN difference Midair Refueling. Ifthesmallestacceptablealtitude the altitudedifference abebetweenthetwo planes? Midair Refueling. Ifthehoseis150feetlong,whatshould should thehosebe? CHAPTER 1 ies 5milesandthenmakes aleftturn(90°)andflies ies 3milesandthenmakes arightturn(90°)andflies a betweenthetwo planesis100feet,how long

Right T riangle Trigonometry

20° 15°

W E 109. 108. 107. 106. 105. Section 1.4 get anerror,explainwhy. Use acalculatortoevaluatethefollowingexpressions.Ifyou

Use acalculatortofind Use acalculatortofind b. a. Calculate cot(34.8°)inthefollowing two ways: b. a. Calculate csc(78.4°)inthefollowing two ways: sec1808 places. 1 bythatnumber. Write thatnumbertofive decimal Find tan(34.8°)tothreedecimalplacesandthendivide by thatnumber. Write thatnumbertofive decimalplaces. Find sin(78.4°)tothreedecimalplacesandthendivide 1 the resulttofive decimalplaces. First findtan(34.8°)andthenitsreciprocal.Round the resulttofive decimalplaces. First findsin(78.4°)andthenitsreciprocal.Round

110.

csc1808 cos tan 3 3 tan cos 2 2 1 1 1 1 2.612 0.125 24 24 . . 15/12/16 3:05 PM PRACTICE TEST

° 0 63

4 15/12/16 3:05 PM

° 5 from a ° . . 7

r r 2 c 9 9 s 7 c 4 3 . Air has an . ° ° 2 5 5 r 5 3 u 1 Practice Test Practice

n 1 2 . i

° s °

r r

r 5 0 If the incidence angle If the incidence angle 0 7 n . 4 3

4 2

. t 0 s b ° ° o 0 b o 5 . 2 2 c c 2 1 1 8 i . u 1 a 1 find

5 , ° i n i º n m 0 s find b.

k 6 , i

40 n n 3 m i 2 , find the exact value of exact , find the c s

2 0 5 1 from degrees-minutes-seconds to to from degrees-minutes-seconds ! s c 5 3 4 1 and the refraction angle in the quartz is and the refraction angle 2

, a r from decimal degrees to degrees-minutes- from decimal degrees 6 ° . and c 5 47 m, find a ° and 0 5 r 7 1 !

2 m 0 2 ° . . k 12 mm, find c. and a 5 12 mm, find and 1

2 ° ° 4 what is the index of refraction of quartz crystal? of refraction of quartz what is the index ° 5 5 , 2 5 2 0 0 4 . 3 i ° ° 1

5 2 9 3 u 6 5 s

, 1 7 o

° 5 5 5 c n 5 5 i 5 2 b s a a a r and point 200 yards from the base of the building. How tall is the How point 200 yards from the base of the building. building? index of refraction value of of refraction value index decimal degrees. Round to the nearest hundredth of a degree. Round decimal degrees. and is flying at an altitude of 150 feet above sea level, what sea level, and is flying at an altitude of 150 feet above on the surface is the diameter of the circle that is illuminated of the water? If If If c 5 14.0 ft, find If b 5 13.3 ft and If If a 5 3.45 and b 5 6.78, find Convert Convert Convert If of value Find the exact Perform the indicated operation: Perform the indicated operation: is to the top of a building The angle of elevation What is the measure (in degrees) of the angle that a second (in degrees) What is the measure hand sweeps in 5 seconds? to quartz crystal appears to bend Light going from air law: according to Snell’s u of helicopter has a field of view If the search and rescue seconds. is 10. 11. 12. 13. 14. 15. 19. 20. 21. 22. 23. 24. 25. 16. 17. 18. 3 ? θ u 6 6 . 1

0 CSC < u

s u o c SEC Round your . 2 and ° 2 3 8 . u b 2 5 4 u u

α 1 u t

n c s o a COT e t c o s

c. f. u a and c TAN PRACTICE TEST] PRACTICE

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answer to four decimal places. appropriate decimal place. Convert Convert Use a calculator to approximate between What is the difference c. below. in the table values Fill in the exact a. d. functions. for the indicated values Find the exact a. b. Find the exact values for the indicated functions. for the indicated values Find the exact In a right triangle, if the side opposite a 30° angle has a In a right triangle, if what is the length of the other leg length of 5 centimeters, and the hypotenuse? and she wants in the Grand Canyon, A 5-foot girl is standing The sun casts (depth) of the canyon. to estimate the height the measure To ground. inches along the 6 shadow her the length she walks cast by the top of the canyon, shadow 200 steps and estimates that each She takes of the shadow. deep is the Grand Approximately how step is roughly 3 feet. Canyon? n n Calculate the measure of three angles in a triangle if the angles in a triangle if measure of three Calculate the are true: following u 308 458 608 CHAPTER 1 CHAPTER For Exercises 10–15, refer to the triangle below: For Exercises 4 and 5, use the triangle below: For Exercises 4 and 5, use the triangle 9. 8. 6. 5. 4. 2. 3. [ 1. 7. Young_Trigonometry_6161_ch01_pp036-063.indd 63