Right Triangle Trigonometry · cause the ball to travel different distances and different heights. One design element of a golf club is the angle of the club face. Sine Ratio Sine

C H A P T E R

7

Chapter 7 | Right Triangle Trigonometry 371

7

7.1 Tangent RatioTangent Ratio, Cotangent Ratio, and

Inverse Tangent | p. 375

7.2 Sine RatioSine Ratio, Cosecant Ratio, and

Inverse Sine | p. 383

7.3 Cosine RatioCosine Ratio, Secant Ratio, and

Inverse Cosine | p. 391

7.4 Angles of Elevation and DepressionAngles of Elevation, Angles of

Depression, and Equivalent

Trigonometric Ratios | p. 399

Golfers use different golf clubs in different circumstances. The angle of the golf

club face determines the path of the ball. Some clubs, such as a wedge, cause

the ball to go very high in the air but travel a short horizontal distance. Other

clubs, such as a driver, cause the ball to stay lower to the ground and travel a long

horizontal distance. You will investigate how club face angles affect the shape of

golf clubs.

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Right Triangle Trigonometry

372 Chapter 7 | Right Triangle Trigonometry

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Introductory Problem for Chapter 7

Wheelchair Ramps

Troy is building wheelchair ramps for his grandfather. He reviewed the Americans

with Disabilities Act (ADA) Accessibility Guidelines to look for wheelchair ramp

specifications. Troy discovered that the maximum incline for a wheelchair ramp should

not exceed a ratio of 1:12. This means that every 1 inch of vertical rise requires

12 inches of horizontal run. The maximum rise for any run is 30 inches. The ability to

manage the incline of the ramp is related to both its steepness and its length.

Troy decides to build a ramp with the ratio 1 : 12.

1. The first ramp extends from the front porch to the front yard. The vertical rise

from the porch to the yard is 2.5 feet.

a. Draw a diagram of the ramp. Include the measurements for the vertical rise

and horizontal run of the ramp.

b. Calculate the surface length of the ramp.

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Chapter 7 | Introductory Problem for Chapter 7 373

2. The second ramp extends from the deck on the back of the house to the back

yard. The vertical rise from the deck to the yard is 18 inches.

a. Draw a diagram of the ramp. Include the measurements for the vertical rise

and horizontal run of the ramp.

b. Calculate the run or horizontal projection of the ramp.

3. Compare the two ramps. Are the triangles similar? Explain.

4. Compare and describe the angles of incline on the two ramps.

Be prepared to share your solutions and methods.

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Lesson 7.1 | Tangent Ratio 375

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In the wheelchair ramp problem, Troy used the rule 1 : 12, the ratio of the rise of the

ramp to the run of the ramp.

1. Describe the shape of the wheelchair ramp problem.

2. What does the ratio of the rise of the ramp to the run of the ramp represent?

3. Consider the right triangle shown. What does the ratio of the rise to the

run represent?

run

rise

PROBLEM 1 Slope and Right Triangles

Tangent RatioTangent Ratio, Cotangent Ratio, and Inverse Tangent

7.1

OBJECTIVESIn this lesson you will:l Use the tangent ratio in a right triangle to solve

for unknown sides.

l Use the cotangent ratio in a right triangle to

solve for unknown sides.

l Relate the cotangent ratio to the tangent ratio.

l Use the inverse tangent to solve for

unknown angles.

KEY TERMSl tangent (tan)

l cotangent (cot)

l inverse tangent

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4. Are the triangles shown similar? Explain your reasoning.

A

B

C

30°9 feet

9�3 feet

D

E

F

30°

24 feet

24�3 feet

a. Calculate the ratio of the rise to the run for each triangle. Write your answers

as fractions in simplest form.

b. How do the ratios compare?

5. Are the triangles below similar? Explain your reasoning.

45°

14 in.

14 in.L

M

N

45°5 in.

5 in.P

Q

R

a. Calculate the ratio of the rise to the run for each triangle. Write your answers

as fractions in simplest form.

b. How do the ratios compare?

6. What can you conclude about the ratios of the rise to the run in similar

right triangles?


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7. In right triangles, you can study the relationship between the interior angles and

the lengths of the sides. Consider �A in the right triangle shown.

a. Which leg of the triangle is opposite �A?

A

B

C

b. Which leg of the triangle is adjacent to �A? In other words, which leg is a

side of �A?

The relationship between �A, the side opposite �A, and the side adjacent to �A

can be defined. The tangent (tan) of an acute angle in a right triangle is the ratio

of the length of the side that is opposite the angle to the length of the side that is

adjacent to the angle.

tan A � length of side opposite �A

___________________________ length of side adjacent to �A

� BC ___ AC

The expression “tan A” means “the tangent of �A.”

8. Complete the ratio that represents the tangent of �B.

tan B � length of side opposite �B

___________________________ length of side adjacent to �B

�

9. Write expressions for the tangents of all the acute angles in the right triangles

from Questions 4 and 5.


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a. What can you conclude about the tangents of congruent angles in

similar triangles?

b. What happens to the tangent value of an angle as the measure of the

angle increases?

c. Consider each of the triangles in Question 4. In each triangle, compare tan

30º to tan 60º. What do you notice?

d. Why do you think this happens?

9. A proposed wheelchair ramp is shown.

4°24 inches

a. What information about the ramp is required to show that the ramp meets

the safety rules?

b. If you calculate the value of tan 4�, how can you use this value to determine

whether the ramp meets the safety rules?

c. Calculate the value of tan 4�. Round your answer to the nearest hundredth.

d. Write a decimal that represents the greatest value of the slope of a safe ramp.

e. What is the ratio of the rise of the ramp to the run of the ramp? Is the

ramp safe?


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10. Another proposed wheelchair ramp is shown. What should the run of the ramp

be so that the ramp meets the safety rules? Use the tangent of an angle to

calculate your answer. If necessary, round your answer to the nearest inch.

4°20 inches

x

11. Another proposed wheelchair ramp is shown. What should the rise of the ramp

be so that the ramp meets the safety rules? Use the tangent of an angle to

calculate your answer. If necessary, round your answer to the nearest inch.

4°

100 inches

x

Check your answer. Show all your work.


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PROBLEM 2 Cotangent RatioThe cotangent (cot) of an acute angle in a right triangle is the ratio of the length

of the side that is adjacent to the angle to the length of the side that is opposite

the angle.

A C

B

1. Complete the ratio that represents the cotangent of �A.

cot A � length of side adjacent to �A

___________________________ length of side opposite �A

� ____

The expression “cot A” represents “the cotangent of �A.”

2. Prove algebraically that the cotangent of A � 1 _____ tan A

.

3. As the measure of an acute angle increases, the tangent of the acute angle

increases. Explain the behavior of the cotangent of an acute angle as the

acute angle increases.

4. If there is no “cot” button on your graphing calculator, how can you compute

the cotangent?


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Use the triangle shown to answer Questions 5 and 6.

5. Solve for x using the cotangent ratio.

6. Solve for x using the tangent ratio.

7. Which ratio was easier to use when calculating the values of x in

Questions 5 and 6? Explain.

B x C

7

21°

A


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The inverse tangent (or arc tangent) of x is defined as the measure of an acute

angle whose tangent is x. If you know the length of any two sides of a right triangle,

it is possible to compute the measure of either acute angle by using the inverse

tangent, or the tan�1 button on your graphing calculator.

In right triangle ABC, if tan A � x, then tan�1 x � m�A.

1. In right triangle ABC, if tan A � 1 __ 5 , calculate tan�1 ( 1 __

5 ) to determine m�A.

2. Determine the ratio for tan B, and then

use tan�1(tan B) to calculate m�B.

15

10

C B

A

3. Calculate m�B. CB

A

3

8

4. Movable bridges are designed to open

waterways for large boats and barges. When

the bridge moves, all vehicle traffic stops.

The maximum height of the open bridge

deck of the movable bridge shown is

37 feet above the water surface.

The waterway width is 85 feet.

Calculate the angle measure formed

by the movement of the bridge.


PROBLEM 3 Inverse Tangent

Lesson 7.2 | Sine Ratio 383

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PROBLEM 1 Fore! Each golf club in a set of clubs is designed to

cause the ball to travel different distances and

different heights. One design element of a golf

club is the angle of the club face.

Sine RatioSine Ratio, Cosecant Ratio, and Inverse Sine

7.2

OBJECTIVESIn this lesson you will:l Use the sine ratio in a right triangle to

solve for unknown sides.l Use the cosecant ratio in a right triangle

to solve for unknown sides.l Relate the cosecant ratio to the sine ratio.l Use the inverse sine to solve for

unknown angles.

KEY TERMSl sine (sin)l cosecant (csc)l inverse sine

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You can draw a right triangle that is formed by the club face angle. The right

triangles formed by different club face angles are shown.

19.3 mm

50.4 mm54 mm

21°

34.4 mm

60 mm 49.2 mm35°

39 mm

62 mm 48.2 mm39°

1. How do you think the club angle affects the path of the ball?

2. For each club face angle, write the ratio of the length of the side opposite the

given acute angle to the length of the hypotenuse. Write your answers as

decimals rounded to the nearest hundredth.

3. What happens to this ratio as the angle measure gets larger?

The sine (sin) of an acute angle in a right triangle is the ratio of the length of the side

that is opposite the angle to the length of the hypotenuse.

A

B

C

4. Complete the ratio that represents the sine of �A.

sin A � length of side opposite �A

_________________________ length of hypotenuse

� ____

The expression “sin A” represents “the sine of �A.”


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5. For each triangle in Problem 1, calculate the value of the sine of the club face

angle. Then calculate the value of the sine of the other acute angle. Round your

answers to the nearest hundredth.

6. For each triangle, do the sine values of the angles appear to be related as the

tangents of the acute angles in a triangle are related?

7. What do the sine values of the angles in Question 5 all have in common?

8. Is the sine of every acute angle less than 1? Explain your reasoning.

9. What happens to the sine values of an angle as the measure of the angle increases?

10. Use the right triangles shown to calculate the values of sin 30�, sin 45�, and sin 60�.

8 feet 4 feet

4�3 feet

30°

A

B

C 8 inches

8 inches8�2 inches

45°

D

E

F

11. A golf club has a club face angle A for which sin A � 0.45. Use your results from

this lesson to estimate the measure of �A. Use a calculator to verify your answer.


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The cosecant (csc) of an acute angle in a right triangle is the ratio of the length of

the hypotenuse to the length of the side that is opposite the angle.

A

B

C

1. Complete the ratio that represents the cosecant of �A.

csc A � length of hypotenuse

_________________________ length of side opposite �A

� ____

The expression “csc A” represents “the cosecant of �A.”

2. Prove algebraically that the cosecant of A � 1 _____ sin A

.

3. As the measure of an acute angle increases, the sine of the acute angle

increases. Explain the behavior of the cosecant of an acute angle as the

acute angle increases.

PROBLEM 2 Cosecant Ratio


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4. If there is no “csc” button on your graphing calculator, how can you compute

the cosecant?

Use the triangle shown to answer Questions 5 and 6.

5. Solve for x using the cosecant ratio.

6. Solve for x using the sine ratio.

B

x

C 12

15°

A


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7. Which ratio was easier to use when calculating the values of x in Questions 5

and 6? Explain.

The inverse sine (or arc sine) of x is defined as the measure of an acute angle whose

sine is x. If you know the length of any two sides of a right triangle, it is possible to

calculate the measure of either acute angle by using the inverse sine,

or sin�1 button on your graphing calculator.

In right triangle ABC, if sin A � x, then sin�1 x � m�A.

1. In right triangle ABC, if sin A � 2 __ 5 , calculate sin�1 ( 2 __


2. Determine the ratio for sin B, and then use sin�1(sin B) to calculate m�B.

C

11

25

A

B

PROBLEM 3 Inverse Sine


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3. Calculate m�B.

C

5

12

A

B

4. This movable bridge shown is called a double-leaf Bascule bridge. It has a

counterweight that continuously balances the bridge deck, or “leaf,”

throughout the entire upward swing, providing an open waterway for boat

traffic. The counterweights on double-leaf bridges are usually located below

the bridge decks.

The length of one leaf, or deck is 42 feet. The maximum height of an open leaf is

30 feet. Calculate the measure of the angle formed by the movement of the bridge.



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Lesson 7.3 | Cosine Ratio 391

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A guy wire is used to provide stability to tall structures like radio towers. Guy wires

are attached near the top of a tower and are attached to the ground.

A guy wire and its tower form a right triangle. It is important that all of the guy wires

form congruent triangles so that the tension on each wire is the same.

1. Do you think that a guy wire provides more support if it is attached to the

ground closer to or farther from the tower?

OBJECTIVESIn this lesson you will:l Use the cosine ratio in a right triangle to

solve for unknown sides.l Use the secant ratio in a right triangle to

solve for unknown sides.l Relate the secant ratio to the cosine ratio.l Use the inverse cosine to solve for

unknown angles.

KEY TERMSl cosine (cos)l trigonometric ratiosl secant (sec)l inverse cosine

7.3

PROBLEM 1 Making Tower Stable

Cosine RatioCosine Ratio, Secant Ratio, and Inverse Cosine

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2. Each triangle shown represents the triangle formed by a tower and guy wire.

The angle formed by the wire and the ground is given in each triangle.

500 ft

300 ft

400 ft

53°

500 ft

335 ft

371 ft

48°

500 ft

250 ft

433 ft

60°

For each angle formed by the wire and the ground, write the ratio of the length

of the side adjacent to the angle to the length of the hypotenuse. Write your

answers as decimals rounded to the nearest hundredth if necessary.

3. What happens to this ratio as the angle measure gets larger?

The cosine (cos) of an acute angle in a right triangle is the ratio of the length of the

side that is adjacent to the angle to the length of the hypotenuse.

A

B

C

4. Complete the ratio that represents the cosine of �A.

cos A � length of side adjacent to �A

___________________________ length of hypotenuse

� ____

The expression “cos A” represents “the cosine of �A.”

5. For each triangle in Question 2, calculate the value of the cosine of the angle

made by the guy wire and the ground. Then calculate the value of the cosine of

the other acute angle. Round your answers to the nearest hundredth

if necessary.


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6. For each triangle, do the cosine values of the angles appear to be related like

the tangents of the acute angles in a triangle are related?

7. What do the cosine values of the angles in Question 6 all have in common?

8. Is the cosine of every acute angle less than 1? Explain your reasoning.

9. What happens to the cosine value of an angle as the measure of the

angle increases?

10. Use the right triangles shown to calculate the values of cos 30º, cos 45º, and

cos 60º. Show all your work.

8 feet

4 feet

30°

A

B C

4�3 feet

8 inches

8 inches8�2 inches

45°D

E

F

11. A guy wire is 600 feet long and forms a 55� angle with the ground. First, draw

a diagram of this situation. Then, calculate the number of feet from the tower's

base to where the wire is attached to the ground.


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The sine, cosine, and tangent of an acute angle are called trigonometric ratios.

12. For the triangle, calculate the values of sin 30�, cos 30�, and tan 30�.

12 meters

6 meters

6�3 meters

30°

A

B

C

13. Calculate the value of sin 30� _______ cos 30�

.

14. What do you notice about the value of sin 30� _______ cos 30�

?

15. Do you think that the relationship between the sine, cosine, and tangent of an

angle is true for any angle? Explain your reasoning.


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The secant (sec) of an acute angle in a right triangle is the ratio of the length of the

hypotenuse to the length of the side that is adjacent to the angle.

A C

B

1. Complete the ratio that represents the secant of �A.

sec A � length of hypotenuse

___________________________ length of side adjacent to �A

� ____

The expression “sec A” represents “the secant of �A.”

2. Prove algebraically that the secant of A � 1 ______ cos A

.

3. As the measure of an acute angle increases, the cosine of the acute angle

decreases. Explain the behavior of the secant of an acute angle as the acute

angle increases.

4. If there is no “sec” button on your graphing calculator, how can you compute

the secant?

PROBLEM 2 Secant Ratio


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Use the triangle shown to answer Question 5 and 6.

5. Solve for x using the secant ratio.

6. Solve for x using the cosine ratio.

7. Which ratio was easier to use when calculating the values of x in Questions 5

and 6? Explain.

A

B

C5

30°

x


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The inverse cosine (or arc cosine) of x is defined as the measure of an acute angle

whose cosine is x. If you know the length of any two sides of a right triangle, it is

possible to compute the measure of either acute angle by using the inverse cosine,

or cos�1 button on your graphing calculator.

In right triangle ABC, if cos A � x, then cos�1 x � m�A.

1. In right triangle ABC, if cos A � 2 __ 7 , calculate cos�1 ( 2 __


2. Determine the ratio for cos B, and then use cos�1(cos B) to calculate m�B.

C

BA 18

16

3. Calculate m�B.

C

B

5

8A

PROBLEM 3 Inverse Cosine


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4. A typical cable-stayed bridge is a continuous girder with one or more towers

erected above piers in the middle of the span. From these towers, cables stretch

down diagonally (usually to both sides) and support the girder. Tension and

compression are calculated into the design of this type of suspension bridge.

One cable is 95 feet. The span on the deck of the bridge from that cable to the

girder is 80 feet. Calculate the angle formed by the deck and the cable.


Lesson 7.4 | Angles of Elevation and Depression 399

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OBJECTIVESIn this lesson you will:l Use trigonometric ratios to solve for the angle

of elevation.l Use trigonometric ratios to solve for the angle

of depression.l Discover equivalent trigonometric ratios.

KEY TERMSl angle of elevationl angle of depression

7.4

PROBLEM 1 Cloud Height The height of the base of clouds is important to weather prediction and aviation. To

measure a cloud’s height from the ground, a ceiling projector and clinometer can be

used. The ceiling projector projects light up vertically toward a cloud and the clinometer

measures the angle of elevation between the ground and the bottom of the cloud.

An angle of elevation is the angle that is formed by a horizontal line and a line from

an observer's eye to a point above the horizontal line.

Angles of Elevation and DepressionAngles of Elevation, Angles of Depression, and Equivalent Trigonometric Ratios

7


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1. A ceiling projector is placed on the ground 1000 feet from a clinometer to

measure the height of the bottom of a cloud mass that is directly above the

projector. The clinometer measures the angle of elevation, or the angle above a

horizontal, to be 81º. Draw a diagram that models this situation.

2. Which of the trigonometric ratios would you use to calculate the vertical

distance from the ground to the bottom of the cloud?

3. Write an equation that you can use to calculate the vertical distance described

in Question 2, and then solve the equation. Round your answer to the nearest

foot if necessary.

4. The ceiling projector is placed 1200 feet from the clinometer to measure the

height of the base of a different cloud mass that is directly above the projector.

The clinometer measures the angle of elevation to be 78º. What is the height of

the base of the clouds? Round your answer to the nearest foot if necessary.


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The clinometer can also measure an angle of depression. An angle of depression

is the angle that is formed by a horizontal line and a line from an observer's eye to a

point below the horizontal line.

5. You are standing on the top of a hill and see a pond below you, as shown in

the diagram. The hill is 200 feet above the pond. The clinometer measures an

angle of depression of 21º to a point directly across from you. Calculate the

horizontal distance between yourself and the pond. Round your answer to the

nearest foot if necessary.

pond

200 feet

you

x

21°

6. You are vacationing at the Grand Canyon. You are standing on the North Rim

of the canyon and you use your clinometer to measure the angle of depression

to a point directly across the canyon on the South Rim. You are at an elevation

of 7256 feet, and the point on the South Rim is at an elevation of 6159 feet.

If you measure the angle of depression to be 19º, what is the width of the

canyon between yourself and the point on the South Rim?


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1. Calculate the six trigonometric ratios for �A from what is given and perform

the same calculations for �B. Round your answers to the nearest thousandths

if necessary.

A

52°

C

B

2. Calculate the six trigonometric ratios for �A from what is given and perform

the same calculations for �B. Round your answers to the nearest thousandths

if necessary.

A6

C

B

8

PROBLEM 2 Exploring Equivalent Trigonometric Ratios


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Use your answers to Questions 1 and 2 to answer Question 3, parts (a) through (f).

3. Describe the relationship between each:

a. The sine of �A and the cosine of �B

b. The cosine of �A and the sine of �B

c. The tangent of �A and the cotangent of �B

d. The cosecant of �A and the secant of �B

e. The secant of �A and the cosecant of �B

f. The cotangent of �A and the tangent of �B

4. Do these equivalent relationships hold true for the measures of any acute

angles in a right triangle? Use the triangle shown and choose your own angle

measurements to determine whether these relationships hold true.

B

A C


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5. List all of the equivalent trigonometric ratios.

6. Based on the relationships in Question 5, why do you think each pair of ratios

were named?

a. Sine and cosine

b. Secant and cosecant

c. Tangent and cotangent


Chapter 7 | Checklist 405

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7Using the Tangent Ratio

The tangent (tan) of an acute angle in a right triangle is the ratio of the length of

the side that is opposite the angle to the length of the side that is adjacent to the

angle. You can use the tangent of an angle to determine the length of a leg in a

right triangle when you know the measure of an acute angle and the length of the

other leg.

Example:

A

B

C

1.5 ft

42°

x

tan 42º � x ___ 1.5

1.5(tan 42º) � x

x � 1.35 ft

Chapter 7 Checklist

KEY TERMSl tangent (tan) (7.1)l cotangent (cot) (7.1)l inverse tangent (7.1)l sine (sin) ratio (7.2)

l cosecant (csc) (7.2)l inverse sine (7.2)l cosine (cos) (7.3)l trigonometric ratios (7.3)

l secant (sec) (7.3)l inverse cosine (7.3)l angle of elevation (7.4)l angle of depression (7.4)

7.1

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Using the Cotangent Ratio

The cotangent (cot) of an acute angle in a right triangle is the ratio of the length of

the side that is adjacent to the angle to the length of the side that is opposite the

angle. You can use the cotangent of an angle to determine the length of a leg in a

right triangle when you know the measure of an acute angle and the length of the

other leg.

Example:

F

G

H

6 m

55°

a

cot 55º � a __ 6

6(cot 55º) � a

6 ( 1 _______

tan 55º ) � a

a � 4.20 m

Using the Inverse Tangent

The inverse tangent, or arc tangent, of x is defined as the measure of an acute angle

whose tangent is x. You can use the inverse tangent to calculate the measure of

either acute angle in a right triangle when you know the lengths of both legs.

Example:

X

Y

5 in.

14 in. Z

m�X � tan�1 ( 5 ___ 14

) � 19.65º

m�Y � tan�1 ( 14 ___ 5

) � 70.35º

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Using the Sine Ratio

The sine (sin) of an acute angle in a right triangle is the ratio of the length of the side

that is opposite the angle to the length of the hypotenuse. You can use the sine of an

angle to determine the length of a leg in a right triangle when you know the measure of

the angle opposite the leg and the length of the hypotenuse. You can also use the sine

of an angle to determine the length of the hypotenuse when you know the measure of

an acute angle and the length of the leg opposite the angle.

Example:

S T

18 cm61°

y

R

sin 61º � y ___

18

18(sin 61º) � y

x � 15.74 cm

Using the Cosecant Ratio

The cosecant (csc) of an acute angle in a right triangle is the ratio of the length of

the hypotenuse to the length of the side that is opposite the angle. You can use the

cosecant of an angle to determine the length of a leg in a right triangle when you know

the measure of the angle opposite the leg and the length of the hypotenuse. You can

also use the cosecant of an angle to determine the length of the hypotenuse when you

know the measure of an acute angle and the length of the leg opposite the angle.

Example:

P Q

10 in.

33°

a

R

csc 33º � a ___ 10

10(csc 33º) � a

10 ( 1 _______

sin 33º ) � a

a � 18.36 in.

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Using the Inverse Sine

The inverse sine, or arc sine, of x is defined as the measure of an acute angle whose

sine is x. You can use the inverse sine to calculate the measure of an acute angle

in a right triangle when you know the length of the leg opposite the angle and the

length of the hypotenuse.

Example:

J

K

44 yd18 yd

L

m�J � sin�1 ( 18 ___ 44

) � 24.15º

Using the Cosine Ratio

The cosine (cos) of an acute angle in a right triangle is the ratio of the length of the

side that is adjacent to the angle to the length of the hypotenuse. You can use the

cosine of an angle to determine the length of a leg in a right triangle when you know

the measure of the angle adjacent to the leg and the length of the hypotenuse.

You can also use the cosine of an angle to determine the length of the hypotenuse

when you know the measure of an acute angle and the length of the leg adjacent to

the angle.

Example:

W

X

9 mm

26°

c

V

cos 26º � c __ 9

9(cos 26º) � c

c � 8.09 mm

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Using the Secant Ratio

The secant (sec) of an acute angle in a right triangle is the ratio of the length of the

hypotenuse to the length of the side that is adjacent to the angle. You can use the

secant of an angle to determine the length of a leg in a right triangle when you know

the measure of the angle adjacent to the leg and the length of the hypotenuse.

You can also use the secant of an angle to determine the length of the hypotenuse

when you know the measure of an acute angle and the length of the leg adjacent

to the angle.

Example:

D

E

25 ft

48°x

F

sec 48º � x ___ 25

25(sec 48º) � x

25 ( 1 _______

cos 48º ) � x

x � 37.36 ft

Using the Inverse Cosine

The inverse cosine, or arc cosine, of x is defined as the measure of an acute angle

whose cosine is x. You can use the inverse cosine to calculate the measure of an

acute angle in a right triangle when you know the length of the leg adjacent to the

angle and the length of the hypotenuse.

Example:

R

M

16 m

7 m

P

m�P � cos�1 ( 7 ___ 16

) � 64.06º

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Using Angles of Elevation

An angle of elevation is the angle above a horizontal. You can use trigonometric

ratios to solve problems involving angles of elevation.

Example:Mitchell is standing on the ground 14 feet from a building and he is looking up at

the top of the building. The angle of elevation that his line of sight makes with the

horizontal is 65°. His eyes are 5.2 feet from the ground. To calculate the height of

the building, first draw a diagram of the situation. Then write and solve an equation

involving a trigonometric ratio.

65°

14 ft 5.2 ft

xx + 5.2 ft

tan 65º � x ___ 14

14(tan 65º) � x

x � 30 ft

x � 5.2 � 30 � 5.2 � 35.2 ft

The building is about 35.2 feet tall.

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Using Angles of Depression

An angle of elevation is the angle below a horizontal. You can use trigonometric

ratios to solve problems involving angles of depression.

Example:You are standing on a cliff and you see a house below you. You are 50 feet above the

house. The angle of depression that your line of sight makes with the horizontal is 33°.

To calculate the horizontal distance x you are from the house, first draw a diagram of

the situation. Then write and solve an equation involving a trigonometric ratio.

33°

57°

50 ft

x

tan 57º � x ___ 50

50(tan 57º) � x

x � 77 ft

You are about 77 feet from the house.

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Documents

Right Triangle Trigonometry · cause the ball to travel different distances and different heights. One design element of a golf club is the angle of the club face. Sine Ratio Sine