Trigonometry for the Solution of Triangles
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Trigonometry was originally a branch of geometry. Its name means triangle measure (from trigos + metry). If two angles of a triangle have measures which are known, then the third angle can be calculated by knowing that the three angles have a sum of 180o. This does not require any knowledge of trigonometry.
mCA B = 38.09 m ACB = 75.23 mFDE = 33.75 F mGEF = 102.25 C
A D E B G
So m < B = ______and m < F = ______
If any two sides of a right triangle have lengths which are known, then the length of the third side requires the use of the Pythagorean Theorem. This also does not require any knowledge of trigonometry.
J K M
3 12.4 45.1
H L 8 I
h2 + j2 = i2 m2 + l2 = k2 32 + 82 = i2 12.42 + l2 = 45.12 so i2 = 73 so l2 = 45.12 – 12.42
i = 73 8.544 l = 1880.25 43.362 When there is a combination of information about angles and sides, then there is a need for additional knowledge. This is where trigonometry comes in. Trigonometry and Triangles page 2
To see how the ideas of trigonometry are developed, we use the ideas of the ratios of the three sides of right triangle. (Later, an extension of these ideas allows trigonometry to be used on triangles which do not have right angles.)
Using the ideas about similar triangles (which are triangles which have the same shapes but different sizes),
F G H
I
E D C B A
all of the right triangles are similar. Thus there are equal ratios for corresponding parts.
Since IB HC GD FE = sin (< A) (by definition, called SOH) AI AH AG AF AB AC AD AE = cos (< A) (by definition, called CAH) AI AH AG AF IB HC GD FE = tan (< A) (by definition, called TOA) AB AC AD AE Trigonometry and Triangles page 3 Let’s draw these again in one triangle so that you can see what the names are for each ratio, using a mnemonic device to help remember which trig names goes with which ratio. hypotenuse opposite leg 90 A adjacent leg a opposite _ leg sine of < A = sin(A) = (called “SOH”) c hypotenuse b adjacent _ leg cosine of < A = cos(A) = (called “CAH”) c hypotenuse a opposite _ leg tangent of < A = tan(A) = (called “TOA”) b adjacent _ leg The mnemonic device some students use to remember the sine, cosine, and tangent ratios is Soh Cah Toa Sine = Opposite / Hypotenuse Cosine = Adjacent / Hypotenuse Tangent = Opposite / Adjacent Now, if you list every possible quotient with the measurements a, b, and c, you should get six possibilities. List all of those six here (e.g., a and c ): b a Trigonometry and Triangles page 4 Three of your possibilities have the names that are listed above. The other three should be the reciprocals of the ones listed above. Their names are cotangent ( = cot (A) or ctn (A) ) for the reciprocal of tangent, secant ( = sec (A) ) for the reciprocal of cosine, and cosecant ( = csc (A) ) for the reciprocal of sine. Fill in these ratios using the information about the other three names described above. b adjacent _ leg cotangent of < A = cot(A) = a opposite _ leg c hypotenuse secant of < A = sec(A) = b adjacent _ leg c hypotenuse cosecant of < A = csc(A) = a opposite _ leg Find the six trigonometric function values of each of A and B . DO not try to calculate the angles yet. 1. B 5 A 12 C 2. A 10 C 6 B Trigonometry and Triangles page 5 Here is how these trig ratios are used to solve right triangles. x 13.2 280 9 12 x x 9 tan 28o = sin x = 12 13.2 9 x = 12 tan 28 x = sin-1 ( ) 13.2 x 6.381 6.4 (two significant digits) x 42.986 43o (to nearest degree) Note 1: Make sure that your calculator is in the DEGREE mode, not the RADIAN mode. (We will discuss radians later.) Note 2: To calculate tan (28o), use the TAN key. To solve sin (B) = 9/13.2 , use the yellow 2nd key, and then sin (which is marked SIN-1 above the SIN key). This is called the “arcsine” or “inverse sine”. Try these. Solve for x (whether x is the length of a side or the measure of an angle). 1.B 2. B 58.1 140 x 0 23 810 A C A X E m Trigonometry and Triangles page 6 3.L M 4. 17.31 B x 12.4 x 17.1 A 8 C K 5. K L x 73.8 27 M Homework: Find the value of x in the right triangles. Round answers to two decimal places. 6. A 125 x B 68 Trigonometry and Triangles page 7 7. 54 x 95 8. 210 22 58 X 9. x 37 63 150 Trigonometry and Triangles page 8 Angles of Elevation and Depression Angle of Elevation Angle of Depression Trigonometry can be used to solve these right triangle problems. 10. To approach runway 17 of the Ponca City Municipal Airport in Oklahoma, the pilot must begin a 3 decent starting from an altitude of 2714 feet. The airport altitude is 1007 feet. How many miles from the runway is the airplane at the start of this approach? 11. Two office buildings are 51 m apart. The height of the taller building is 207 m. The angle of elevation from the top of the shorter building to the top of the taller building is 15. Find the height of the shorter building. Trigonometry and Triangles page 9 12. You are standing on the top of a 200 foot building watching a car drive toward the building. The car stops at a traffic light and you measure the angle of depression to the car to be 13.8°. The car then drives closer to the building and stops at the next traffic light, where you measure the angle of depression to the car to be 35.5°. How far apart are the two traffic lights? Homework: Angle of Elevation and Depression Problems Make a diagram, label it and solve. Do work neatly. Give answers to one place after the decimal. 13. The angle of elevation of the top of the Louisiana State Capital from a point on the ground 829 feet from its base is 28.5. Find the height of the building. 14. A balloon is 539 feet above one end of a bridge that spans the Mississippi River at Vicksburg, Mississippi. The angle of depression of the other end of the bridge from the balloon is 33.2. How long is the bridge? Trigonometry and Triangles page 10 15. From the top of a building across the street from a skyscraper, the angle of depression of the bottom of the skyscraper is 38.9 and the angle of elevation of the top of the skyscraper is 69.4. If the buildings are 150 feet apart, find the height of the skyscraper. 16. To find the height of a building, you start by measuring the angle of elevation to the top of the building to be 73°. You then walk 20 feet closer to the building and measure the angle of elevation to be 78°. What is the height of the building? 17. From the top of a lighthouse 180 feet above sea level at high tide, the angle of depression of a buoy is 34.17 at high tide and 34.50 at low tide. Find the height of the tide. (That is, how far does the water drops from high tide to low tide.) Trigonometry and Triangles page 11 18. When the angle of elevation of the sun is 42.5°, a tree casts a shadow that is 15.6 feet long. Find the height of the tree. 19. The sun rose at 6:15 a.m. and it will be directly overhead at 12:15 p.m. What time is it when a 24 foot pole casts a 17 foot shadow? Something to experiment with: Use your calculator to see what happens when you evaluate these. Make sure you are in DEGREE MODE. a. (SIN(26))^2 +(COS(26))^2 b. (SIN(51))^2 +(COS(51))^2 c. (SIN(167))^2 +(COS(167))^2 d. (SIN(6))^2 +(COS(6))^2 e. (SIN(90))^2 +(COS(90))^2 f. Using inductive thinking, generalize to conclude that sin2 x + cos2 x = ______. Trigonometry and Triangles page 12 Law of Cosines 20. Can you use trigonometry to find the length of the missing side? Explain. 20 35 24 21. Trigonometry WITHOUT right triangles – the Law of Cosines Proof: 22. For what types of problems can the Law of Cosines be used? Trigonometry and Triangles page 13 Exercises with the law of cosines. Find the missing side. 23. 15.78 127.4 2.65 24. Find the measure x and y. y 30 26 x 46 Trigonometry and Triangles page 14 Homework: Law of Cosines Each problem refers to ABC in which side a is opposite A, side b is opposite B, and side c is opposite C. Solve for the indicated part of the triangle. Give answers to two decimal places. 25. a 12, b 14, c 20 Find m C 26. B 72.3, a 78, c 16 Find b. 27. a 26.12, b 21.34, c 19.25 Find m A Trigonometry and Triangles page 15 28. C 28.43, a 6, b 9 Find c. 29. The longer base of an isosceles trapezoid measures 14 ft. The nonparallel sides measure 10 ft., and the base angles measure 80. Find the length of a diagonal. 30. Three circles are arranged as shown in the figure below. Find the length of PQ. 1.1 P 109 1.4cm 1.8cm Q 31. A weather balloon is directly west of two observation stations that are 10 miles apart. The angles of elevation of the balloon from the two stations are 17.6 and 78.2. How high is the balloon? Law of Sines What is the Law of Sines? In any triangle, the sides are proportional to the sines of the opposite angles. Proof: 32. For what types of problems can the Law of Sines be used? Trigonometry and Triangles page 17 33. Find the missing parts of each triangle. a) 125 122 18 b) 48 98 73 Homework: Law of Sines Find all the missing parts of each triangle. Round lengths to the nearest tenth, and angles to the nearest tenth of a degree. 34. 152 110 28 Trigonometry and Triangles page 18 35. 250 15 120 36. 52 38 47 37. 45 43 58 Trigonometry and Triangles page 19 38. The Leaning Tower of Pisa makes an angle of 5.33 with the vertical. When the angle of elevation of the sun is 36.33, the shadow of the tower (falling on the horizontal plane) is 227 feet long. Find the distance (x) from the bottom to the top of the tower. x 36.33 shadow 5.33 39. Find the length of x rounded to hundredths. 30 40 5 x 60 25 35 55 Trigonometry and Triangles page 20 40. Find AB. B A 33 42 41 C 37 100 cm D The Ambiguous Case – Geometer’s Sketchpad Investigation The “Ambiguous Case” refers to problems in which the given information about a triangle is ______. For this investigation we will assume we are given angle A, side a, and side b. GOAL: to construct as many triangles as possible with the given information in order to develop some rules for determining how many triangles can be formed. Use Geometer’s Sketchpad. Trigonometry and Triangles page 21 Angle A a < b a > b a = b Acute Obtuse Use the conjectures you reached above to answer this question. How many triangles can be constructed with the given information? 41. A = 30, a = 6, and b = 7 ______ 42. A = 48, a = 24, and b = 20 ______ 43. A = 84, a = 15, and b = 58 ______ 44. A = 124, a = 40, and b = 40 ______ 45. B = 74, a = 68, and b = 70 ______ Trigonometry and Triangles page 22 Solve for all missing parts (lengths and angle measures) of each triangle. 46. 10 6 28 47. A 24 , a = 2.4. and b = 3.1 Trigonometry and Triangles page 23 48. A 138 , a = 42.65. and b = 26.95 Homework: Ambiguous Case Find all the missing parts of each triangle. Round answers to one place after the decimal. If more than one triangle can be formed, solve both triangles. I no triangle can be formed, explain why. 49. A 65.2 , a = 125.9, and b = 130.2 Trigonometry and Triangles page 24 50. A 110 , a = 45, and b = 51 51. A 87 , a = 645, and b = 720 52. A 10.5 , a =100, and b = 148 Trigonometry and Triangles page 25 53. Observers are located on the side of a hill at points P and Q. The hill is inclined 32 with the horizontal. The observer at P determines the angle of elevation to a hot-air balloon to be 62. At the same time, the observer at Q measures the angle of elevation to the balloon to be 71. If P is 60 m down the hill from Q, find the distance from Q to the balloon. 54. Two lighthouses (A and B) on a straight shoreline are 120 miles apart. At the same time, an observer at each lighthouse spots a ship cruising along the coastline. The observer at A measures the angle between the shoreline and the ship to be 42.3. The observer at B measures the angle between the shoreline and the ship to be 68.9. Find the shortest distance from the ship to the shore. ship B A Trigonometry and Triangles page 26 55. A tree is growing vertically on a hillside. The angle of inclination of the hill is 22 with the horizontal. When the angle of elevation of the sun is 52, the tree casts a shadow 215 feet down the hill. Find the height of the tree. 22