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Unit 6

Trigonometry

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Lesson: Sine, Cosine, and Tangent For any acute angle in a right triangle, we denote the measure of the angle by θ (theta) and define ratios related to θ as follows: sin 𝜃 = cos 𝜃 = tan 𝜃 = Guided Practice:

a. Use the Pythagorean Theorem to find x, the measure of the leg

opposite 𝜃.

b. Find the values for the three trigonometric ratios for angle 𝜃.

Often, you will be given the measure of 𝜃. YOU MUST MAKE SURE YOUR CALCULATE IS IN DEGREE MODE! If you are using any of the TI-83/84 calculators:

Press MODE

Use the arrow keys to highlight “Degree”

Press ENTER once degree is highlighted in black

Press 2nd, MODE to return to the home screen

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We will now practice finding a missing side length given and one known side length. Remember to label the opposite, adjacent, and hypotenuse sides to determine which trigonometric ratio to use.

1. Use an appropriate ratio to write and solve an equation to find the missing side length. Round to the tenths place.

(a) x 51o 4 (b) 21 65o x (c) x 78o 4 (d) 9 x 24o

4

18

x

27°

25

x

36°x

16

41°

x

7

52°

x

6

25°

8

41 °

x

65°

14

x

Skills Practice using SOH CAH TOA

Find the missing side of the triangle. Show all work. Round answers to the nearest tenth. 1. 2. 3. 4. 5. 6. 7. 8.

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Additional Practice

Find the value of each trigonometric ratio.

Find the missing side. Round to the nearest tenth.

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Find the area of each triangle. Round all answers to the nearest tenth.

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Answers to Additional Practice:

Learning Task: Finding the Missing Angle using SOH CAH TOA

1. An airport is tracking the path of one of its incoming flights. The distance to the plane is 850 ft. (from the ground) and the altitude of the plane is 400 ft.

(a) What is the sine ratio of the angle of elevation from the ground at the airport to the plane (refer to the figure)?

(b) What is the cosine ratio of the angle of elevation? (Hint: use the Pythagorean Theorem to solve for the missing side of the right triangle before setting up the ratio)

(c) What is the tangent ratio of the angle of elevation?

(d) Now, use your calculator to find the measure of the angle itself. Pressing “2nd” followed by one of the trigonometric function keys finds the angle measure corresponding to a given ratio.

Press 2nd SIN Type the sine ratio from part a. What value do you get? Round your answer to two decimal places.

(e) Press 2nd COS Type in the cosine ratio from part b. What value do you get?

8

9

5

α7

24

β

(f) Press 2nd TAN Type in the tangent ratio from part c. What value do you get?

(g) Why did you get the same answer each time?

(h) To the nearest hundredth of a degree, what is the measure of the angle of elevation?

Guided Practice: 1. The top of a billboard is 40 feet above the ground. What is the angle of elevation of the sun when the billboard casts a 30-foot shadow on level ground?

a. Sketch a figure to illustrate the problem.

b. Use the inverse trigonometric function to the find the angle of elevation. Round to the nearest hundredth of a degree.

For problems 2-5, find the missing angle. Round to the nearest hundredth of a degree. 2. 3.

Did you notice that, for each of the calculations in parts d – f, when you pressed 2nd , sin, the calculator typed sin-1. This means that we are taking the inverse of the function. This allows us to find the angle given two sides of the triangle.

9

3

5

α 6

17

β

15

α11.3

α24

12 α

14

7

5 4

ββ

2528

β3

9

4. 5.

Skills Practice using Inverse Trigonometric Functions Solve for the missing degree measure. Round to the nearest hundredth of a degree. 1. 2. 3. α = ____________ α = ____________ α = ____________ 4. 5. 6. β = ____________ β = ____________ β = ____________

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Additional Practice:

Answers to Additional Practice:

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Applications of Trigonometry 1. Brian and Ben are playing golf and both of their golf balls have

landed on a flat portion of the green. The distance, d, between

Ben's golf ball and Brian's golf ball is 5 feet. If 𝛼 = 60°, then

how far is Ben's golf ball from the center of the hole? Round

your answer to the nearest foot.

2. The manager of a gas station has attached a cable with flags on it to a light pole in order to attract more business. The cable is attached 15 feet above the base of the light and forms a 50° angle at the ground. Find the length of the cable, r. Round your answer to the nearest tenth of a foot.

3. One afternoon, a tree casts a shadow that is 35.6

feet long. At that time, the angle of elevation of

the sun is 45°, as shown in the figure at the right.

How tall is the tree? Round your answer to the

nearest foot.

Note that an angle of elevation is measured up from the horizontal because, to look up at something, you need to raise, or elevate, your line of sight from the horizontal.

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4. Daniel sees a lighthouse in the harbor. He estimates the angle of elevation is 70°. If the lighthouse is 120 feet tall, what is the approximate distance between Daniel and the top of the lighthouse? (Assume the lighthouse meets the ground at a right angle.) Round your answer to the nearest foot.

5. A forest ranger is on a fire lookout tower in a national forest. His observation position is 214.7 feet above the ground when he spots an illegal campfire. The angle of depression of the line of site to the campfire is 12°.

(a) The angle of depression is equal to the corresponding angle of elevation. Why?

(b) Assuming that the ground is level, how far is it from the base of the tower to the campfire?

Note that an angle of depression is measured down from the horizontal because, to look down at something, you need to lower, or depress, your line of sight from the horizontal.

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6. Ricardo is standing 75 feet away from the base of a building. The angle of elevation from the ground where Ricardo is standing to the top of the building is 32°.

What is x, the height of the building, to the nearest tenth of a foot? 7. An airplane is at an altitude of 5,900 feet. The airplane descends at an angle of 3°.

a. Explain why the angle of depression is congruent to the angle of elevation.

b. About how far will the airplane travel in the air until it reaches the ground? Round your answer the nearest foot.

8. A ramp with an elevation of 11⁰ leads to a door that has a brick base that is 45 feet away from the start of the ramp. How many feet long is the ramp? Round your answer to the nearest foot.

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9. From a 200 feet high cliff a boat is noticed floundering at sea! The boat is approximately 300 yards from the base of the cliff. What is the angle of depression, to the nearest degree, of the line of sight to the boat?

10. A bird rises 20 meters vertically over a horizontal distance of 80 meters. What is the angle of elevation? Round your answer to the nearest tenth. 11. A ladder is leaning against the side of a house so that the distance on the ground between the base of the ladder and the house is 7 feet. If the length of the ladder is 15 feet, then what is the angle at which the ladder is leaning? At what height does it reach the house? Round your answer to the nearest tenth. 12. Find, to the nearest degree, the angle which the sun’s rays make with the ground when a flagpole 40 feet high casts a shadow 30 feet long.

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13. During its approach to Earth, the space shuttle’s glide angle changes. When the shuttle’s altitude is about 15.7 miles, its horizontal distance to the runway is about 59 miles. What is its glide angle? Round your answer to the nearest tenth.

Skills Practice Applications of Trigonometry

1. A ladder is leaning against a building. The ladder is 10m long and it is sitting on the ground 4m out

from the building. What is the angle that the ladder makes with the ground?

2. A sailboat’s main sail is shaped like a right triangle and the base is 18m long. If it makes an angle of

60, as marked, how tall is the sail?

3. The Leaning Tower of Pisa is 55m tall. The top edge of the tower is 5m out from the bottom edge.

What is the angle created between the ground and the tower?

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4. After a windstorm, one of the hydro poles had a lean to it. The poles are 14m high and the angle

between the pole and the ground is 81. How high is the top of the pole above the ground? Be sure to draw a figure first.

5. A 6.1 meter ladder leans against a wall. The angle formed by the ladder and the ground is 71.

a. Draw a figure that represents the situation.

b. How far is the foot of the ladder from the wall?

c. How high up the wall does the ladder reach?

6. A kite has a string 200m long. The string makes an angle of 43 with the ground. Determine the height of the kite.

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7. An 8.5 foot tall snow plough covers a distance of 14.1 feet across the ground. At what angle is the snow plough ploughing up snow from the ground?

8. A guy wire is 15 meters long. It supports a vertical television tower. The wire is fastened to the

ground 9.6m out from the base of the tower.

a. Calculate the angle formed by the guy wire and the ground.

b. Calculate how far up the tower the guy wire attaches.

9. A truck travels 8km up a mountain road. The change in height from the bottom to the top is 1400m.

Find the angle of inclination of the road.

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10. A roller coaster climbs vertically 60metres at an angle of 40 from the lowest to the highest point of the track. It then plunges over the high point to begin the ‘fun part’. Calculate the length of the track from the bottom of the hill to the very top.

11. A window in an apartment building is 32m above the ground. From the window, the angle of elevation

of the top of an apartment building across the street is 36. The angle of depression to the bottom of

the same apartment building is 47. Determine the height of the building across the street.

12. A tree casts a 23m shadow when the angle of elevation of the sun is 52.

a. Find the height of the tree.

b. Find the length of the shadow when the angle of elevation of the sun is 38.

13. From the top of a cliff, which is 120 meters above the water, the angel of depression of a boat on the

water is 18. How far is the boat from the bottom of the cliff?

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C

A B

Investigation Task - Complements

You will need: a ruler, a scientific calculator In this investigation you will make a small table of trigonometric ratios for angles measuring 20° and 70°.

Step 1 The right triangle ABC, below, has angle measures of: 20 , 90 , 70m A m B m C

Step 2 Measure AB, AC, and BC to the nearest millimeter. Write the lengths on your triangle. Step 3 Use your side lengths and the definitions of sine, cosine, and tangent to complete the table. Round your calculations to the nearest thousandth. Often, you will be given the measure of 𝜃. YOU MUST MAKE SURE YOUR CALCULATE IS IN DEGREE MODE! If you are using any of the TI-83/84 calculators:

Press MODE

Use the arrow keys to highlight “Degree”

Press ENTER once degree is highlighted in black

Press 2nd, MODE to return to the home screen

Step 4 Share your results with your group. What observations can you make about the trigonometric ratios you found? What is the relationship between the values for 20° and the values for 70°? Explain why you think these relationships exist. Today, trigonometric tables have been replaced by calculators that have sin, cos, and tan keys. Step 5 Experiment with your calculator to determine how to find the sine, cosine, and tangent values of angles. Step 6 Use your calculator to find sin 20°, cos 20°, tan 20°, sin 70°, cos 70°, and tan 70°. Check your group’s table. How do the trigonometric ratios found by measuring sides compare with the trigonometric ratios you found on the calculator?

mA sin A cos A tan A

20

mC sin C cos C tan C

70

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Step 7 Recall that the acute angles of a right triangle are complementary angles. Would the same relationship for the trigonometric ratios that you found in Step 6 exist with other complementary angles? Test your conjecture by repeating Step 6 with right triangles with the following acute angle measures:

DEF with 40m D , 50m F

RST with 30m R , 60m T

XYZ with 45m X , 45m Z

Fill in the missing angle for the text box:

In any right triangle: sin = ______________

cos = ______________

sin(90 ) = ______________

cos(90 ) ______________

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Guided Practice:

1. Let cos θ =2

5 . What is the sin (90-θ)⁰?

2. Let sin θ =6

10 . What is the cos(90-θ)⁰? (Sketch a picture of the triangle)

3a. Write the trigonometric function for represented in the right triangle below. b. What is the length of the missing leg of the triangle? Find the following values: cos =________ tan =________

sin =________ tan =________ sin

cos

=________

4. Given tan = 7

24, draw a right triangle that would represent this trigonometric ratio.

Find the following:

sin ___________

sin ____________ sin(90 ) __________

cos(90 ) ___________

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5

22

θ

1237

θ

Skills Practice:

1. Given sin = 8

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cos = ______________ sin (90- ) = ___________ cos (90- ) = ___________ 2. Given the following trigonometric values, label the triangle’s sides.

tan = 9

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cos = __________ sin(90- ) = _________ cos(90- ) = __________ sin = ___________ 3. Given the triangle below, find the length missing side. Then answer the questions about the triangle.

Missing side length = ____________ sin = __________ cos = __________ sin(90- ) = __________ cos(90- ) = ____________

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How do I know what to do when solving a right triangle?

Right Triangles

Solving for a Side Solving for an Angle

OR

Given 2 sides Given 1 side AND 1 angle

Given 2 sides Given 2 angles

Pythagorean Theorem

SOHCAHTOA Inverse SOHCAHTOA

Triangle Sum Theorem

Ratios

Set up with SOHCAHTOA

Use Complement

Rules