Download 10.4 Tangent Ratio - Schurz High School

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10.4
Goal
Find the tangent of an
acute angle.
Key Words
• trigonometric ratio
• leg opposite an angle
Tangent Ratio
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 trigonometric ratio
is a ratio of the lengths of two sides of a
right triangle.
• leg adjacent to an angle
• tangent
leg
opposite
angle
leg adjacent
For any acute angle of a right triangle,
there is a leg opposite the angle and
a leg adjacent to the angle. The ratio
of these legs is the tangent of the angle.
TANGENT RATIO
For any acute angle A of a right triangle:
B
hypotenuse
Student Help
READING TIP
leg opposite aA
a
tan A b
leg adjacent to aA
The tangent of angle A
is written as “tan A.”
A
EXAMPLE
1
leg
a opposite
aA
c
b
leg adjacent to aA
C
Find Tangent Ratio
Find tan S and tan R as fractions in simplified form
and as decimals rounded to four decimal places.
R
Solution
leg opposite aS
leg adjacent to aS
43
4
leg opposite aR
leg adjacent to aR
4
43
S
8
43
4
T
tan S 3
≈ 1.7321
1
3
tan R ≈ 0.5774
10.4
Tangent Ratio
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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.
Student Help
EXAMPLE
KEYSTROKE HELP
When calculating the
tangent of an angle, be
sure your calculator is
in DEGREE mode.
Use a Calculator for Tangent
2
Approximate tan 74 to four decimal places.
Solution
Calculator keystrokes
74
or
74
Display
Rounded value
3.487414444
3.4874
Find Tangent Ratio
Find tan S and tan R as fractions in simplified form and as decimals.
Round to four decimal places if necessary.
1.
10
S
8
R
2. R
6
10
T
T
26
24
S
Use a calculator to approximate the value to four decimal places.
3. tan 35
EXAMPLE
4. tan 85
5. tan 10
Find Leg Length
3
Use a tangent ratio to find the value of x.
Round your answer to the nearest tenth.
3
22
x
Solution
opposite leg
adjacent leg
Write the tangent ratio.
3
x
Substitute.
tan 22 tan 22 x p tan 22 3
3
tan 22
x Chapter 10
Divide each side by tan 22.
x ≈ Use a calculator or table
to approximate tan 22.
x ≈ 7.4
Simplify.
3
0.4040
558
Multiply each side by x.
Right Triangles and Trigonometry
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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.
Student Help
STUDY TIP
Given the measure of
one acute angle of a
right triangle, use the
Corollary to the Triangle
Sum Theorem to find
the measure of the
other angle.
EXAMPLE
Find Leg Length
4
Use two different tangent ratios to find the value
of x to the nearest tenth.
4
35
x
Solution
First, find the measure of the other acute angle: 90 35 55.
Method 1
opposite leg
tan 35 adjacent leg
Method 2
opposite leg
tan 55 adjacent leg
x
4
tan 35 4
x
tan 55 x p tan 35 4
4 tan 55 x
4
tan 35
4(1.4281) ≈ x
x 4
0.7002
x ≈ x ≈ 5.7
x ≈ 5.7
ANSWER
The two methods yield the same answer: x ≈ 5.7.
Careers
EXAMPLE
Estimate Height
5
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.
h
Solution
FORESTERS Foresters
manage and protect forests.
To determine the height of a
tree, a forester may use
trigonometry.
tan 59 opposite leg
adjacent leg
Write ratio.
h
45
Substitute.
tan 59 45 tan 59 h
Multiply each side by 45.
45(1.6643) ≈ h
Use a calculator or table to
approximate tan 59.
74.9 ≈ h
Career Links
CLASSZONE.COM
ANSWER
Simplify.
The tree is about 75 feet tall.
10.4
Tangent Ratio
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Find Side Length
Write two equations you can use to find the value of x.
6.
7.
8.
x 44
4
5
37
59
x
x
8
Find the value of x. Round your answer to the nearest tenth.
9.
10.
11.
18
20 60
7
x
34
x
55
x
10.4 Exercises
Guided Practice
Vocabulary Check
1. Name the acute angles in T DEF.
E
2. Identify the leg opposite aD and the
D
leg adjacent to aD.
Skill Check
F
Find tan A as a fraction in simplified form and as a decimal. Round to
four decimal places if necessary.
3. A
9
C
4.
15
12
5. A
A
52
5
83
C
8
16
B
B
B
5
C
Use a calculator to approximate the value to four decimal places.
6. tan 25
7. tan 62
8. tan 80
9. tan 43
Practice and Applications
Extra Practice
Finding Tangent Ratios Find tan A. Write your answer as a fraction.
See p. 693.
10. B
8
C
560
Chapter 10
11.
A
17
15
Right Triangles and Trigonometry
12
A
B
63
6
C
12. C
2
B
3
13
A
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Homework Help
Example 1: Exs. 10–15
Example 2: Exs. 16–23
Example 3: Exs. 24–29,
32–37
Example 4: Exs. 24–29,
32–37
Example 5: Exs. 31, 38
Finding Tangent Ratios Find tan P and tan R as fractions in
simplified form and as decimals rounded to four decimal places.
13. P
24
14.
25
37
R
12
35
P
P
15. P
15
20
P
25
R
P 7 R
Using a Calculator Use a calculator to approximate the value to four
decimal places.
16. tan 28
17. tan 54
18. tan 5
19. tan 89
20. tan 67
21. tan 40
22. tan 12
23. tan 83
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.
35
x 39
9
x
26.
56
x
70
33
Finding Leg Lengths Find the value of x to the nearest tenth.
27.
x
28.
29.
20
29
52
12
53
41
x
x
30. Error Analysis To find the length
B
&* in the diagram at the right,
of BC
Water Slides
18
BC
a student wrote tan 55 .
Explain the student’s error.
A
18
30
55
C
31. Water Slide A water slide makes an angle of about 13
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.
Not drawn to scale
WATER SLIDES Riders on
some of the newer water
slides may travel at speeds
greater than 50 mi/h.
Application Links
58.2 meters
CLASSZONE.COM
10.4
Tangent Ratio
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Finding Side Lengths Find the value of x to the nearest tenth.
IStudent Help
ICLASSZONE.COM
32.
HOMEWORK HELP
Extra help with problem
solving in Exs. 32–37 is
at classzone.com
33.
x
49
34.
x
x
70
34
10
14
3
35.
35
36.
x
37.
20
x
45
x
40
35
50
10
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.
Standardized Test
Practice
39. Multiple Choice Which expression can be used to find the value
of x in the triangle shown?
A
x 10 tan 38
10
C x tan 38
tan 38
B x 10
10
10
D x tan 52
38
x
40. Multiple Choice What is the approximate value of y in the
triangle shown?
F
H
Mixed Review
G
J
7.2
9.3
8.4
50 6
10.1
y
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.
6m
43.
14 in.
8 ft
9m
4 ft
Algebra Skills
562
Chapter 10
10 in.
5 ft
Solving Equations Solve the equation. (Skills Review, p. 673)
44. 8x 10 3x
45. 4(x 3) 32
46. 3x 7 x 11
47. 6x 5 3x 4
48. 2 x 4x 22
49. 5x 18 2x 21
Right Triangles and Trigonometry