How can you find the height of the tree in

the photograph at the right? It is too tall to

be measured directly in any simple manner.

You can determine the height using a

trigonometric ratio. A

is a ratio of the lengths of two sides of a

right triangle.

For any acute angle of a right triangle,

there is a the angle and

a to the angle. The ratio

of these legs is the of the angle.tangent

leg adjacent

leg opposite

trigonometric ratio

10.4 Tangent Ratio 557

Goal Find the tangent of an

acute angle.

Key Words• trigonometric ratio

• leg opposite an angle

• leg adjacent to an angle

• tangent

10.410.4 Tangent Ratio

READING TIP

The tangent of angle Ais written as “tan A.”

Student Help

Find tan S and tan R as fractions in simplified form

and as decimals rounded to four decimal places.

Solution

tan S 5 5 5 Ï3w ≈ 1.7321

tan R 5 5 5 ≈ 0.57741

}Ï3w

4}4Ï3w

leg opposite aR}}}leg adjacent to aR

4Ï3w}

4

leg opposite aS}}}leg adjacent to aS

EXAMPLE 1 Find Tangent Ratio

R T

S

8

4Ï·3

4

For any acute angle A of a right triangle:

tan A 5 5 }b

a}

leg opposite aA}}}leg adjacent to aA

TANGENT RATIO

b

c

A

hypotenuse

leg adjacent to aA

legoppositeaA

a

C

B

leg opposite

angle

leg adjacent

Page 1 of 6

Tangent Function You can use the TAN function on a calculator to

approximate the tangent of an angle. You can also use the table of

trigonometric ratios on page 705.

Find tan S and tan R as fractions in simplified form and as decimals.

Round to four decimal places if necessary.

1. 2.

Use a calculator to approximate the value to four decimal places.

3. tan 358 4. tan 858 5. tan 108

10

T

R

S24

2610

S T

R

8

6

558 Chapter 10 Right Triangles and Trigonometry

Approximate tan 748 to four decimal places.

Solution

Calculator keystrokes Display Rounded value

74 or 74 3.487414444 3.4874

EXAMPLE 2 Use a Calculator for Tangent

Use a tangent ratio to find the value of x.

Round your answer to the nearest tenth.

Solution

tan 228 5 Write the tangent ratio.

tan 228 5 }3x

} Substitute.

x p tan 228 5 3 Multiply each side by x.

x 5 }tan

3228} Divide each side by tan 228.

x ≈ }0.4

3040}

x ≈ 7.4 Simplify.

Use a calculator or table

to approximate tan 228.

opposite leg}}adjacent leg

EXAMPLE 3 Find Leg Length

Find Tangent Ratio

228

x

3

KEYSTROKE HELP

When calculating thetangent of an angle, besure your calculator isin DEGREE mode.

Student Help

Page 2 of 6

In Example 3, the unknown, x, was in the denominator of the ratio.

If you prefer to use a ratio in which the unknown is in the numerator,

use the tangent ratio for the other acute angle in the triangle.

10.4 Tangent Ratio 559

Use two different tangent ratios to find the value

of x to the nearest tenth.

Solution

First, find the measure of the other acute angle: 908 2 358 5 558.

Method 1 Method 2

tan 358 5 tan 558 5

tan 358 5 }4x

} tan 558 5 }4x

}

x p tan 358 5 4 4 tan 558 5 x

x 5 }tan

4358} 4(1.4281) ≈ x

x ≈ }0.7

4002} x ≈ 5.7

x ≈ 5.7

ANSWER © The two methods yield the same answer: x ≈ 5.7.

opposite leg}}adjacent leg

opposite leg}}adjacent leg

EXAMPLE 4 Find Leg Length

You stand 45 feet from the base of a tree and

look up at the top of the tree as shown in the

diagram. Use a tangent ratio to estimate the

height of the tree to the nearest foot.

Solution

tan 598 5 Write ratio.

tan 598 5 }4h5} Substitute.

45 tan 598 5 h Multiply each side by 45.

45(1.6643) ≈ h

74.9 ≈ h Simplify.

ANSWER © The tree is about 75 feet tall.

Use a calculator or table toapproximate tan 598.

opposite leg}}adjacent leg

EXAMPLE 5 Estimate Height

358

x

4

h

STUDY TIP

Given the measure ofone acute angle of aright triangle, use theCorollary to the TriangleSum Theorem to findthe measure of theother angle.

Student Help

FORESTERS Foresters

manage and protect forests.

To determine the height of a

tree, a forester may use

trigonometry.

Career LinksC L A S S Z O N E . C O M

Careers

Page 3 of 6

Write two equations you can use to find the value of x.

6. 7. 8.

Find the value of x. Round your answer to the nearest tenth.

9. 10. 11.

60820

x558

x

18

348

7

x

5

598

x

378

x

4448x

8

560 Chapter 10 Right Triangles and Trigonometry

Find Side Length

1. Name the acute angles in TDEF.

2. Identify the leg opposite aD and the

leg adjacent to aD.

Find tan A as a fraction in simplified form and as a decimal. Round to

four decimal places if necessary.

3. 4. 5.

Use a calculator to approximate the value to four decimal places.

6. tan 258 7. tan 628 8. tan 808 9. tan 438

Finding Tangent Ratios Find tan A. Write your answer as a fraction.

10. 11. 12. A3

2Ï·13·

B

C

126

A

6Ï·3B C15

817

A

B

C

Practice and Applications

8

8Ï·3

16

A

B

C

5

55Ï·2

A

B C

9

12

15

A

BC

Skill Check

Vocabulary Check

Guided Practice

Exercises10.410.4

D

E

F

Extra Practice

See p. 693.

Page 4 of 6

10.4 Tangent Ratio 561

Finding Tangent Ratios Find tan P and tan R as fractions in

simplified form and as decimals rounded to four decimal places.

13. 14. 15.

Using a Calculator Use a calculator to approximate the value to four

decimal places.

16. tan 288 17. tan 548 18. tan 58 19. tan 898

20. tan 678 21. tan 408 22. tan 128 23. tan 838

Using Tangent Ratios Write two equations you can use to find the

value of x. Then find the value of x to the nearest tenth.

24. 25. 26.

Finding Leg Lengths Find the value of x to the nearest tenth.

27. 28. 29.

30. Error Analysis To find the length

of BC&* in the diagram at the right,

a student wrote tan 558 5 }B1

C8} .

Explain the student’s error.

31. Water Slide A water slide makes an angle of about 138

with the ground. The slide extends horizontally about

58.2 meters as shown below. Find the height h of the slide

to the nearest tenth of a meter.

538x

20298

x

1252

418

x

358

x

70398x

33

9

568

x

15

R

P P

2520

37R

PP 35

12

24

P

P

R7

25

308 558

18

A

B

C

Example 1: Exs. 10–15Example 2: Exs. 16–23Example 3: Exs. 24–29,

32–37Example 4: Exs. 24–29,

32–37Example 5: Exs. 31, 38

Homework Help

58.2 meters

Not drawn to scale

WATER SLIDES Riders on

some of the newer water

slides may travel at speeds

greater than 50 mi/h.

Application LinksC L A S S Z O N E . C O M

Water Slides

Page 5 of 6

562 Chapter 10 Right Triangles and Trigonometry

38. Surveying To find the distance d

from a house on shore to a house on

an island, a surveyor measures from

the house on shore to point B, as

shown in the diagram. An instrument

called a transit is used to find the

measure of aB. Find the distance d

to the nearest tenth of a meter.

39. Multiple Choice Which expression can be used to find the value

of x in the triangle shown?

XA x 5 10 tan 388 XB x 5 }tan

10388}

XC x 5 }tan

10388} XD x 5 }

tan10

528}

40. Multiple Choice What is the approximate value of y in the

triangle shown?

XF 7.2 XG 8.4

XH 9.3 XJ 10.1

Volume of Solids Find the volume of the solid. If necessary, round

your answer to the nearest whole number. (Lessons 9.4, 9.5)

41. 42. 43.

Solving Equations Solve the equation. (Skills Review, p. 673)

44. 8x 2 10 5 3x 45. 4(x 1 3) 5 32 46. 3x 2 7 2 x 5 11

47. 6x 1 5 5 3x 2 4 48. 2 2 x 5 4x 1 22 49. 5x 2 18 5 2x 1 21

Algebra Skills

10 in.

14 in.

8 ft

5 ft4 ft

9 m

6 m

Mixed Review

Standardized TestPractice

388

10

x

508 6

y

Finding Side Lengths Find the value of x to the nearest tenth.

32. 33. 34.

35. 36. 37.

458 508

10x

358

20x

408

35

x

348

10

x498

14

x

708

3

x

HOMEWORK HELP

Extra help with problem

solving in Exs. 32–37 is

at classzone.com

IStudent HelpI C L A S S Z O N E . C O M

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