Trigonometry Packet Geometry Honors

Hertel/Williams/Lambert

Trigonometry packet Geometry honors

Table of contents:

Day 1: Basic Trigonometry Review SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles.

Day 2: Trig Review and Co-Functions SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction.

Day 3: Using Trigonometry to Determine Area SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known.

Day 4: Law of Sines SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles.

Day 5: Law of Cosines SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle.

Day 6: Review

Day 7: TEST

SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles.

Day 1

Basic Trigonometry Review

Warm Up: Review the basic Trig Rules below and complete the example below:

Basic Trigonometry Rules:

 These formulas ONLY work in a right triangle.

 The hypotenuse is across from the right angle.

 Questions usually ask for an answer to the nearest units.

 You need a scientific or graphing calculator.

Example:

SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles. Example 1:

Practice:

1.)

SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles.

2.)

Example 2:

Example 3: Inverse Trigonometric Relations to find Missing Angles.

SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles.

Practice:

3.)

4.)

5.)

SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles.

Exit Ticket:

Summary:

SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles.

Day 1 Homework:

SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles.

8. 9.

10. 11.

SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction.

Day 2 Basic Trig Word Problems and CoFunctions

Warm Up: 1.) Find the length of side p and the measure of angle m, as shown on the diagram. Give each answer correct to the nearest whole number or degree.

2.)

SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction.

Real World Connection: Trigonometry can be used on a daily basis in the workplace. Since trigonometry means "triangle measure", any profession that deals with measurement deals with trigonometry as well. Carpenters, construction workers and engineers, for example, must possess a thorough understanding of trigonometry.

Angle of Elevation The angle of elevation is always measured from the ground up. Think of it like an elevator that only goes up. It is always INSIDE the triangle.

In the diagram at the left, x marks the angle of elevation of the top of the tree as seen from a point on the ground.

You can think of the angle of elevation in relation to the movement of your eyes. You are looking straight ahead and you must raise (elevate) your eyes to see the top of the tree.

Angle of Depression The angle of depression is always OUTSIDE the triangle. It is never inside the triangle.

In the diagram at the left, x marks the angle of depression of a boat at sea from the top of a lighthouse.

You can think of the angle of depression in relation to the movement of your eyes. You are standing at the top of the lighthouse and you are looking straight ahead. You must lower (depress) your eyes to see the boat in the water.

There are two possible ways to use our angle of depression to obtain an angle INSIDE the triangle.

1. Find the angle adjacent (next door) to our angle which is inside the triangle. This adjacent angle will always be the complement of our angle. Our angle and the angle next door will add to 90º. In the diagram on the left, the adjacent angle is 55º. 2. Utilize the fact that the angle of depression = the angle of elevation and simply place 35º in angle A. (the easiest method) Just be sure to place it in the proper position.

SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction.

Example 1:

Practice:

1.) From a point on the ground 25 feet from the foot of a tree, the angle of elevation of the top of the tree is 32º. Find to the nearest foot, the height of the tree.

2.)

SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction.

3.)

Example 2: Refer to the triangle below:

a) What is the relationship between mA and mB? ______A 13 12 b) What is the cos A? _____ What is the sine B? _____

C 5 B c) What is the sin A? _____ What is the cos B? _____

What do you notice about the cosine and sine of complements?

______

The sine of an acute angle is equal to the cosine of its complement.

The sine and cosine functions are called cofunctions.

SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction.

Any trigonometric function of an acute angle is equal to the cofunction of its complement.

cos x = sin (90o - x) sin x = cos (90o - x)

o o Forsec  example, = csc (90 - find) the value csc of  = sin sec 30(90o. - )

Now find the value of cos 60o.

SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction.

Exit Ticket:

Summary:

SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction.

Homework:

1. Find the value of that makes each statement true. a. ( ) b. ( ) c. ( )

d. ( )

2. a. Make a prediction about how the sum will relate to the sum . b. Use the sine and cosine values of special angles to find the sum: . c. Find the sum . d. Was your prediction a valid prediction? Explain why or why not.

3. Langdon thinks that the sum is equal to . Do you agree with Langdon? Explain what this means about the sum of the sines of angles.

4.)

5.)

6.)

SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction.

Summary: SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction.

SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known.

Day 3 Using Trigonometry to Determine Area

Warm Up: Determine the Area for each triangle below, if possible. If not possible, state what additional information would be needed.

Triangle #1 Triangle #2 Triangle #3

*If there is missing information, is there a way to find the missing information?

SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known. Example 1: What if the third side length of the triangle were provided? Is it possible to determine the area of the triangle now? Find the area of . (Hint: Draw Altitude HJ)

SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known. Discussion: Let’s look at which is set up similarly to the triangle in Example 1:

What formula would represent the area of this triangle? A= ½ ah

Let’s rewrite the equation. Describe what you see:

A= ½ ah ______

A= ½ ab ______

Look at below with an arc mark at vertex labeled . What do you notice about the structure of ? Can we think of this newly written area formula in a different way using trigonometry? Area of a Triangle:

(Circle the part that represents the height)

Using the new formula, what information would you need to calculate the area of a triangle?

SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known. Example 2: A farmer is planning how to divide his land for planting next year’s crops. A triangular plot of land is left with two known side lengths measuring and . How could the farmer calculate the area of the plot, if the included angle between the known side lengths is 30⁰?

You Try It: A real estate developer and her surveyor are searching for their next piece of land to build on. They each examine a plot of land in the shape of . The real estate developer measures the length of and and finds them to both be approximately feet, and the included angle has a measure of approximately . The surveyor measures the length of and and finds the lengths to be approximately feet and feet, respectively, and measures the angle between the two sides to be approximately . a. Draw a diagram that models the situation, labeling all lengths and angle measures.

b. The real estate developer and surveyor each calculate the area of the plot of land and both find roughly the same area. Show how each person calculated the area; round to the nearest hundred.

SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known. Challenge: In isosceles triangle , the base , and the base angles have measures of . Find the area of .

Summary:

We have just discovered that the area of a triangle can be expressed using the lengths of two sides and the sine of the included angle. This is often referred to as the SAS Formula for the area of a triangle.

The "letters" in the formula may change from problem to problem, so try to remember the pattern of "two sides and the sine of the included angle".

We no longer have to rely on a problem supplying us with the length of the altitude (height) of the triangle in order for us to find the area of the triangle. If we know two sides and the included angle, we are in business.

SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known. Exit Ticket:

Given two sides of the triangle shown, having lengths of and , and their included angle of , find the area of the triangle.

Homework:

Find the area of each triangle. Round each answer to the nearest tenth. 1.

SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known.

5. In , , , and . Determine the area of the triangle. Round to the nearest tenth.

6. A landscape designer is designing a flower garden for a triangular area that is bounded on two sides by the client’s house and driveway. The length of the edges of the garden along the house and driveway are and respectively, and the edges come together at an angle of . Draw a diagram, and then find the area of the garden to the nearest square foot.

7. A right rectangular pyramid has a square base with sides of length . Each lateral face of the pyramid is an isosceles triangle. The angle on each lateral face between the base of the triangle and the adjacent edge is . Find the surface area of the pyramid to the nearest tenth.

8. The Pentagon Building in Washington D.C. is built in the shape of a regular pentagon. Each side of the pentagon measures in length. The building has a pentagonal courtyard with the same center. Each wall of the center courtyard has a length of . What is the approximate area of the roof of the Pentagon Building?

SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known.

9. A regular hexagon is inscribed in a circle with a radius of . Find the perimeter and area of the hexagon.

10. In the figure below, is acute. Show that .

11. Let be a quadrilateral. Let be the measure of the acute angle formed by diagonals ̅̅̅ ̅ and ̅ ̅̅ ̅. Show that

(Hint: Apply the result from Problem 10 to and .)

SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles.

Day 4 Law of Sines

Warm Up: 1.) Find the lengths of and .

2.) Find the lengths of and . How is this different from part (a)?

Law of Sines: Today we will show how to find unknown measurements in triangles that are not right triangles; we can use these formulas for acute and obtuse triangles, but today we will specifically study acute triangles. The two formulas are the Law of Sines and the Law of Cosines. Today we will work with the Law of Sines.

. LAW OF SINES: For ALL TRINGLES. Given with angles , , and and the sides opposite them , , and , the Law of Sines states:

SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles.

The ratio of the sine of an angle in a triangle to the side opposite the angle is the same for each angle in the triangle. *The objective of being able to state the law in words is to move the focus away from specific letters and generalize the formula for any situation.

Proof: Consider with an altitude drawn from to .

a) What is Sin ? o Sin = o h =

b) What is Sin ? o Sin = o h =

How can we use the information above to prove a part of the Law of Sines?

SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles. Example 1: A surveyor needs to determine the distance between two points and that lie on opposite banks of a river. A point is chosen meters from point , on the same side of the river as . The measures of angles and are and , respectively. Approximate the distance from to to the nearest meter. Use the Law of Sines to set up all possible ratios.

Practice:

1.) In , , , and . Find . Include a diagram in your answer.

2.) A car is moving towards a tunnel carved out of the base of a hill. As the accompanying diagram shows, the top of the hill, , is sighted from two locations, and . The distance between and is . What is the height, , of the hill to the nearest foot?

SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles. More Practice: 3.)

4.) Solve for x:

Challenge:

SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles. Summary:

Exit Ticket: In the accompanying diagram of , , , and .

What is the length of side to the nearest tenth? 1) 6.6 3) 11.5 2) 10.1 4) 12.0

Homework:

1. Given , , , and , calculate the measure of angle to the nearest tenth of a degree, and use the Law of Sines to find the lengths of and to the nearest tenth.

Calculate the area of to the nearest square unit.

2. Given , , and , calculate the measure of , and use the Law of Sines to find the lengths of ̅̅̅ ̅ and ̅ ̅̅ ̅ to the nearest hundredth. SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles.

3. Does the law of sines apply to a right triangle? Based on , the following ratios were set up according to the law of sines.

Fill in the partially completed work below.

What conclusions can we draw?

4. Given quadrilateral , , , , is a right angle and , use the law of sines to find the length of , and then find the lengths of ̅̅̅ and ̅̅ ̅ to the nearest tenth of an inch.

5. To find the distance across a canyon, a surveying team locates points A and B on one side of the canyon and point C on the other side of the canyon. The distance between points A and B is 85 yards. The measure of  SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles.

CAB = 68°, and the measure of  ABC = 75°. What is the distance across the canyon?

6. A surveyor walks 380 meters from point A to point B. Then the surveyor turns 80° to the left and walks 240 meters to point C, as in the figure below. What is the approximate length of the marsh?

SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle.

Day 5 Law of Cosines

Warm Up: A man is standing 100 feet away from the building, he notices that the angle of elevation to the top of the building is 41º and that the angle of elevation to a poster on the side of the building is 21º. How far is the poster from the roof of the building?

Law of Cosines: In the last lesson we explored one formula that showed us how to find unknown measurements in triangles that are not right triangles. Today we will learn another one, called the Law of Cosines.

 LAW OF COSINES: For ALL TRINGLES. with angles , , and and the sides opposite them , , and , the law of cosines states:

The square of one side of the triangle is equal to the sum of the squares of the other two sides minus the twice the product of the other two sides and the cosine of the angle between them.

*The objective of being able to state the law in words is to move the focus away from specific letters and generalize the formula for any situation. SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle. The Law of Cosines is really just the Pythagorean Theorem, with a something extra so it works for all triangles. Let’s test this out.

Try it when the  What happens to the right side of the equation?

______

Let’s look back at from the last lesson, with an altitude drawn from to , but this time the point where the altitude meets , point , divides into lengths and .

Express e and h using Trigonometry with respect to .

o h =

o e =

SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle.

Now we turn to right triangle . What length relationship can be concluded between the sides of the triangle?

 By the Pythagorean theorem, the length relationship in is

Substitute the trigonometric expressions for and into this statement. (b = e + d)

After simplifying this equation as much as possible we end up with:

Other arrangements of this formula:  

SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle. Example 1: Our friend the surveyor from yesterday is doing some further work. He has already found the distance between points and (132 m). Now he wants to locate a point that is equidistant from both and and on the same side of the river as . He has his assistant mark the point so that the angles and both measure . What is the distance between and to the nearest meter?

Practice 1: Parallelogram has sides of lengths and , and one of the angles has a measure of . Approximate the length of diagonal to the nearest .

SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle. Challenge:

Summary:

Exit Ticket:

SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle. Homework:

1. Given , use the law of cosines to find the length of the side marked to the nearest tenth.

2. Given triangle , , , and , use the law of cosines to find the length of ̅̅̅ ̅̅ to the nearest tenth.

3. Given triangle , , , and . Draw a diagram of triangle , and use the law of cosines to find the length of ̅̅̅ ̅.

Calculate the area of triangle .

SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle.