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10-6 Trigonometric Ratios
Find the values of the three trigonometric
ratios for angle A .
3. 1. SOLUTION: SOLUTION: 4. SOLUTION: 2. SOLUTION: CCSS TOOLS Use a calculator to find the value
of each trigonometric ratio to the nearest tenthousandth.
5. sin 37°
SOLUTION: 3. SOLUTION: eSolutions Manual - Powered by Cognero
Keystrokes:
37
Then, sin 37° = 0.6018. 6. cos 23°
SOLUTION: Page 1
SOLUTION: : Ratios
37
10-6Keystrokes
Trigonometric
Then, sin 37° = 0.6018. 6. cos 23°
SOLUTION: 10. Keystrokes:
23
Then, cos 23° = 0.9205
SOLUTION: Find the measure of
7. tan 14°
.
SOLUTION: Keystrokes:
14
Then, tan 14° = 0.2493.
Find
.
8. cos 82°
SOLUTION: Keystrokes:
82
Then, cos 82° = 0.1392.
Find
.
Solve each right triangle. Round each side
length to the nearest tenth.
9. SOLUTION: Find the measure of
.
SOLUTION: Find the measure of
Find
11. .
.
Find
.
Find
.
Find
.
10. SOLUTION: Find the measure of
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.
Page 2
10-6 Trigonometric Ratios
The length of the run of the hill is about 11,326.2 ft.
Find m X for each right triangle to the nearest
degree.
12. SOLUTION: Find the measure of
14. .
SOLUTION: –1
Use tan
Find
on a calculator.
Keystrokes:
.
-1
[TAN ]
Find
.
15. SOLUTION: 13. SNOWBOARDING A hill used for snowboarding
has a vertical drop of 3500 feet. The angle the run
makes with the ground is 18°. Estimate the length of r.
–1
Use cos
on a calculator.
Keystrokes:
-1
[COS ]
SOLUTION: The length of the run of the hill is about 11,326.2 ft.
Find m X for each right triangle to the nearest
degree.
16. SOLUTION: –1
Use tan
on a calculator.
Keystrokes:
-1
[TAN ]
14. SOLUTION: –1
Use Manual
tan on
a calculator.
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Keystrokes:
-1
[TAN ]
Page 3
Keystrokes:
Keystrokes:
[TAN ]
[SIN ]
10-6 Trigonometric Ratios
Find the values of the three trigonometric
ratios for angle B.
17. SOLUTION: Use sin
–1
18. on a calculator.
Keystrokes:
SOLUTION: Find
-1
[SIN ]
.
Find the values of the three trigonometric
ratios for angle B.
18. SOLUTION: Find
.
19. SOLUTION: Find
.
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10-6 Trigonometric Ratios
19. 20. SOLUTION: Find
.
SOLUTION: Find
.
CCSS TOOLS Use a calculator to find the value
of each trigonometric ratio to the nearest tenthousandth.
21. tan 2°
20. SOLUTION: SOLUTION: Find
Keystrokes:
2
Then, tan 2° = 0.0349.
.
22. sin 89°
SOLUTION: Keystrokes:
89
Then, sin 89° = 0.9998.
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23. cos 44°
SOLUTION: 28. tan 60°
SOLUTION: SOLUTION: Keystrokes:
2
10-6Then, tan 2° = 0.0349.
Trigonometric Ratios
Keystrokes:
60
Then, tan 60° = 1.7321.
22. sin 89°
Solve each right triangle. Round each side
length to the nearest tenth.
SOLUTION: Keystrokes:
89
Then, sin 89° = 0.9998.
23. cos 44°
29. SOLUTION: SOLUTION: Find the measure of
Keystrokes:
44
Then, cos 44° = 0.7193.
24. tan 45°
Find
SOLUTION: Keystrokes:
Then, tan 45° = 1.
.
.
45
25. sin 73°
SOLUTION: Keystrokes:
73
Then, sin 73° = 0.9563
Find
.
26. cos 90°
SOLUTION: Keystrokes:
Then, cos 90° = 0.
90
27. sin 30°
30. SOLUTION: SOLUTION: Find the measure of
Keystrokes:
30
Then, sin 30° = 0.5.
.
28. tan 60°
SOLUTION: Find
.
Keystrokes:
60
Then, tan 60° = 1.7321.
Solve each right triangle. Round each side
length to the nearest tenth.
Find
.
29. SOLUTION: Find the measure of
.
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31. 10-6 Trigonometric Ratios
31. SOLUTION: Find the measure of
.
33. SOLUTION: Find the measure of
Find
.
.
Find
.
Find
.
Find
.
32. SOLUTION: Find the measure of
.
34. SOLUTION: Find the measure of
.
Find
.
Find
.
Find
.
Find
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.
Page 7
35. ESCALATORS At a local mall, an escalator is 110
feet long. The angle the escalator makes with the
10-6 Trigonometric Ratios
The escalator is about 53 ft high.
Find m J for each right triangle to the nearest
degree.
34. SOLUTION: Find the measure of
36. .
SOLUTION: You know the measure of the side opposite ∠J and
the measure of the hypotenuse. Use the sine ratio.
Find
.
–1
Use a calculator and the [sin ]function to find the
measure of the angle.
Keystrokes:
Find
.
-1
[SIN ] 10
24
24.62431835
So, m∠J ≈ 25°.
35. ESCALATORS At a local mall, an escalator is 110
feet long. The angle the escalator makes with the
ground is 29°. Find the height reached by the escalator.
37. SOLUTION: SOLUTION: You know the measure of the side opposite ∠J and
the measure of the hypotenuse. Use the sine ratio.
–1
Use a calculator and the [sin ] function to find the
measure of the angle.
The escalator is about 53 ft high.
Find m J for each right triangle to the nearest
degree.
Keystrokes:
-1
[SIN ] 15
17
61.92751306
So, m∠J ≈ 62°.
36. SOLUTION: eSolutions
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YouManual
know -the
measure
of the side opposite ∠J and
the measure of the hypotenuse. Use the sine ratio.
Page 8
61.92751306
So, m∠J ≈ 62°.
10-6 Trigonometric Ratios
30.96375653
So, m∠J ≈ 31°.
40. SOLUTION: You know the measure of the side adjacent to ∠J
and the measure of the hypotenuse. Use the cosine
ratio.
38. SOLUTION: You know the measure of the side opposite ∠J and
the measure of the side adjacent to ∠J. Use the
tangent ratio.
–1
Use a calculator and the [cos ] function to find the
measure of the angle.
–1
Use a calculator and the [tan ] function to find the
measure of the angle.
Keystrokes:
Keystrokes:
-1
[TAN ] 23
-1
[COS ] 5
16
71.79004314
14
So, m∠J ≈ 72°.
58.67130713
So, m∠J ≈ 59°.
41. 39. SOLUTION: You know the measure of the side opposite ∠J and
the measure of the side adjacent to ∠J. Use the
tangent ratio.
SOLUTION: You know the measure of the side adjacent to ∠J
and the measure of the hypotenuse. Use the cosine
ratio.
–1
Use a calculator and the [tan ] function to find the
measure of the angle.
–1
Use a calculator and the [cos ] function to find the
measure of the angle.
Keystrokes:
-1
[TAN ] 6
10
30.96375653
Keystrokes:
-1
[COS ] 11
So, m∠J ≈ 31°.
49.67978493
So, m∠J ≈ 50°.
40. SOLUTION: eSolutions
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You know the measure of the side adjacent to ∠J
and the measure of the hypotenuse. Use the cosine
17 42. MONUMENTS The Lincoln Memorial building
measures 204 feet long, 134 feet wide, and 99 feet
tall. Chloe is looking at the top of the monument at an
angle of 55°. How far away is she standing from the monument?
Page 9
SOLUTION: 49.67978493
m∠J ≈ 50°. Ratios
10-6So,
Trigonometric
42. MONUMENTS The Lincoln Memorial building
measures 204 feet long, 134 feet wide, and 99 feet
tall. Chloe is looking at the top of the monument at an
angle of 55°. How far away is she standing from the monument?
SOLUTION: The horizontal distance to the city is about 35,577 ft.
44. FORESTS A forest ranger estimates the height of
a tree is about 175 feet. If the forest ranger is
standing 100 feet from the base of the tree, what is
the measure of the angle formed between the range
and the top of the tree?
SOLUTION: The angle formed between the ground and the top of
the tree is about 60°.
Chloe is standing about 69 ft away from the
monument.
43. AIRPLANES Ella looks down at a city from an
airplane window. The airplane is 5000 feet in the air,
and she looks down at an angle of 8°. Determine the horizontal distance to the city.
Suppose
ABC .
A is an acute angle of right triangle
45. Find sin A and tan A if cos
.
SOLUTION: SOLUTION: The horizontal distance to the city is about 35,577 ft.
44. FORESTS A forest ranger estimates the height of
a tree is about 175 feet. If the forest ranger is
standing 100 feet from the base of the tree, what is
the measure of the angle formed between the range
and the top of the tree?
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SOLUTION: 46. Find tan A and cos A if sin
SOLUTION: .
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10-6 Trigonometric Ratios
46. Find tan A and cos A if sin
.
.
SOLUTION: SOLUTION: 47. Find cos A and tan A if sin
48. Find sin A and cos A if tan
.
SOLUTION: 49. SUBMARINES A submarine descends into the
ocean at an angle of 10° below the water line and travels 3 miles diagonally. How far beneath the
surface of the water has the submarine reached?
SOLUTION: The submarine went about 0.5 mi beneath the
surface of the water.
48. Find sin A and cos A if tan
.
50. MULTIPLE REPRESENTATIONS In this
problem, you will explore a relationship between the
sine and cosine functions.
SOLUTION: a. TABULAR Copy and complete the table using
the triangles shown above.
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b. VERBAL Make a conjecture about the sum of
10-6 Trigonometric Ratios
a. TABULAR Copy and complete the table using
the triangles shown above.
b. The sum of the squares of the sine and cosine of
an acute angle in a right triangle is equal to 1.
51. CHALLENGE Find a and c in the triangle shown.
b. VERBAL Make a conjecture about the sum of
the squares of the sine and cosine of an acute angle
of an acute angle in a right triangle.
SOLUTION: a.
SOLUTION: Use the sum of the angles of a triangle to determine
the value of a.
Use the value of a to determine the value of the
angles of the triangle.
The angles are 90°, 6(5) – 3 or 27°, and 12(5) + 3 or 63°.
Use a trigonometric ratio to find the value of c.
Therefore, a = 5 and c ≈ 7.3.
Then ,
52. REASONING Use the definitions of the sine and
cosine ratios to define the tangent ratio.
SOLUTION: Sin is defined as
and Cos is defined as
Tan can be defined as
.
because:
b. The sum of the squares of the sine and cosine of
an acute angle in a right triangle is equal to 1.
51. CHALLENGE Find a and c in the triangle shown.
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Therefore, a = 5 and c ≈ 7.3.
10-6 Trigonometric Ratios
52. REASONING Use the definitions of the sine and
cosine ratios to define the tangent ratio.
SOLUTION: Sin is defined as
and Cos is defined as
Tan can be defined as
.
54. CCSS ARGUMENTS The sine and cosine of an
acute angle in a right triangle are equal. What can
you conclude about the triangle?
SOLUTION: Given: ΔABC with sides a, b, and c as shown;
sin A = cos A
because:
53. WRITING IN MATH How can triangles be used
to solve problems?
SOLUTION: Many real world problems involve trying to determine
the correct height or length of a given structure.
When lengths and angles are known, right triangles
can be drawn, and trigonometric ratios can be used
to determine missing sides and angles. Similarly, other situations may require triangles and
the Pythagorean theorem to determine unknown
lengths. If a = b, then
. A triangle that has two
congruent sides is called an isosceles triangle.
Therefore, this triangle is an isosceles right triangle.
The legs of the right triangle are equal to each other.
55. WRITING IN MATH Explain how to use
trigonometric ratios to find the missing length of a
side of a right triangle given the measure of one
acute angle and the length of one side.
SOLUTION: Use the acute angle given and the measure of the
known side to set up one of the trigonometric ratios.
The sine ratio uses the opposite side and hypotenuse
of the triangle. The cosine ratio uses the adjacent
side and hypotenuse of the triangle. The tangent ratio
uses the opposite and adjacent sides of the triangle.
Choose the ratio that can be used to solve for the
unknown measure.
Given the following triangle find the missing sides a
and c.
54. CCSS ARGUMENTS The sine and cosine of an
acute angle in a right triangle are equal. What can
you conclude about the triangle?
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SOLUTION: Given: ΔABC with sides a, b, and c as shown;
sin A = cos A
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Since you know the measure of ∠A, set up the
uses the opposite and adjacent sides of the triangle.
Choose the ratio that can be used to solve for the
unknown measure.
10-6 Trigonometric Ratios
Given the following triangle find the missing sides a
and c.
56. Which graph below represents the solution set for −2
≤ x ≤ 4?
A
B
C
Since you know the measure of ∠A, set up the
trigonometric ratios for the acute angle of 42°.
Let a be the measure of the side opposite ∠A, 15 is
the measure of the side adjacent ∠A, and c is the
measure of the hypotenuse.
So, if you are trying to find the measure of a, use
the tangent ratio. If you are trying to find the
measure of c, use the cosine ratio.
D
SOLUTION: The inequality uses less than or equal to signs, so the
points on the graph must be solid. So, choices B and
D are incorrect. In the inequality, x is found between
the two values, so choice C in incorrect.
The correct choice is A.
57. PROBABILITY Suppose one chip is chosen from a bin with the chips shown. To the nearest tenth,
what is the probability that a green chip is chosen?
F 0.2
G 0.5
H 0.6
J 0.8
56. Which graph below represents the solution set for −2
≤ x ≤ 4?
A
SOLUTION: B
C
The correct choice is F.
58. In the graph, for what value(s) of x is y = 0?
D
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The inequality uses less than or equal to signs, so the
points on the graph must be solid. So, choices B and
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10-6 Trigonometric Ratios
The correct choice is F.
a. Let h represent the height reached by the ladder.
Use the Pythagorean Theorem to represent the
value of h in terms of the other two sides.
58. In the graph, for what value(s) of x is y = 0?
If the bottom of the ladder is moved closer to the
base of the house, the distance the bottom of the
2
A 0
B −1
C 1
D 1 and −1
SOLUTION: The graph crosses the x-axis twice, so there are two
values of x for which y = 0. Choices A, B, and C
only offer one x value.
The correct choice is D.
59. EXTENDED RESPONSE A 16-foot ladder is
placed against the side of a house so that the bottom
of the ladder is 8 feet from the base of the house.
a. If the bottom of the ladder is moved closer to the
base of the house, does the height reached by the
ladder increase or decrease?
b. What conclusion can you make about the distance
between the bottom of the ladder and the base of the
house and the height reached by the ladder?
c. How high does the ladder reach if the ladder is 3
feet from the base of the house?
ladder is from the wall will decrease. When 16 is
2
subtracted by a number smaller than 8 , the
2
2
difference is greater than 16 - 8 . Since you are
finding the square root of a larger number, h will be
greater. Therefore, as the bottom of the ladder is
moved closer to the base of the house, the height
reached by the ladder will increase.
b. Sample answer: Let h represent the height
reached by the ladder and d represent the distance
between the bottom of the ladder and the base of the
house. The house is built perpendicular to the ground,
so the ladder will form a right triangle when it is
placed against the side of the house. Use the
Pythagorean Theorem to relate the sides of the
triangle.
2
Therefore, the sum of their squares is 16 or 256.
c.
SOLUTION: If the ladder is 3 ft from the base of the house, then it
reaches a height of about 15.7 ft.
If c is the measure of the hypotenuse of a right
triangle, find each missing measure. If
necessary, round to the nearest hundredth.
60. a = 16, b = 63, c = ?
SOLUTION: a. Let h represent the height reached by the ladder.
Use the Pythagorean Theorem to represent the
value of h in terms of the other two sides.
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The length of the hypotenuse is 65 units.
If the bottom of the ladder is moved closer to the
The length of one of the legs is
units.
the ladder is 3 ftRatios
from the base of the house, then it
10-6IfTrigonometric
reaches a height of about 15.7 ft.
If c is the measure of the hypotenuse of a right
triangle, find each missing measure. If
necessary, round to the nearest hundredth.
60. a = 16, b = 63, c = ?
or about 10.72
63. a = 6, b = 3, c = ?
SOLUTION: SOLUTION: The length of the hypotenuse is
units.
The length of the hypotenuse is 65 units.
64. 61. b = 3,
,c=?
or about 6.71 , c = 12, a = ?
SOLUTION: SOLUTION: The length of the hypotenuse is 11 units.
The length of one of the legs is
units.
or about 8.19 62. c = 14, a = 9, b = ?
65. a = 4,
SOLUTION: The length of one of the legs is
units.
63. a = 6, b = 3, c = ?
SOLUTION: ,c=?
SOLUTION: or about 10.72
The length of the hypotenuse is
units.
or about 5.20 66. AVIATION The relationship between a plane’s
length L in feet and the pounds P its wings can lift is
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described by
, where k is the constant of
proportionality. A Boeing 747 is 232 feet long and
has a takeoff weight of 870,000 pounds. Determine k
Page 16
for this plane to the nearest hundredth.
SOLUTION: length of the hypotenuse
is
10-6The
Trigonometric
Ratios
units.
or about 5.20 66. AVIATION The relationship between a plane’s
length L in feet and the pounds P its wings can lift is
69. described by
, where k is the constant of
proportionality. A Boeing 747 is 232 feet long and
has a takeoff weight of 870,000 pounds. Determine k
for this plane to the nearest hundredth.
SOLUTION: SOLUTION: 70. The constant of proportionality is about 0.06.
SOLUTION: 67. FINANCIAL LITERACY A salesperson is paid
$32,000 a year plus 5% of the amount in sales made.
What is the amount of sales needed to have an
annual income greater than $45,000?
SOLUTION: Let x represent the amount of sales made.
71. The amount of sales must be more than $260,000.
Solve each proportion.
SOLUTION: 68. SOLUTION: 69. SOLUTION: eSolutions
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