Download 4-1 Right Triangle Trigonometry

4-1 Right Triangle Trigonometry
Find the exact values of the six trigonometric functions of θ.
2. ANSWER: sin θ = , cos θ = , tan θ = , csc θ = , sec θ = , cot θ = 4. ANSWER: sin θ = , cos θ = , tan θ = , csc θ = , sec θ = , cot θ = 6. ANSWER: sin θ = , cos θ = , tan θ = , csc θ = , sec θ = , cot θ = 8. ANSWER: sin θ = , cos θ = , tan θ = , csc θ = , sec θ = , cot θ = 4
Use the given trigonometric function value of the acute angle θ to find the exact values of the five
remaining trigonometric function values of θ.
10. cos θ = ANSWER: sin θ = 12. sec θ = 8
, tan θ = eSolutions Manual - Powered by Cognero
ANSWER: , csc θ = , sec θ = , cot θ = Page 1
ANSWER: sin θ = , tan θ = , csc θ = 4-1 Right
Triangle
Trigonometry
, sec θ = , cot θ = 12. sec θ = 8
ANSWER: sin θ = , cos θ = , tan θ = 3
, csc θ = , cot θ = 14. tan θ = ANSWER: sin θ = , cos θ = , csc θ = , sec θ = , cot θ = 4
16. csc θ = 6
ANSWER: sin θ = , cos θ = , tan θ = , sec θ = , cot θ = 18. sin θ = ANSWER: cos θ = , tan θ = , csc θ = , sec θ = , cot θ = Find the value of x. Round to the nearest tenth, if necessary.
20. ANSWER: 6.5
22. ANSWER: 17.5
24. ANSWER: 22.2
eSolutions Manual - Powered by Cognero
26. Page 2
22. ANSWER: 4-1 Right
Triangle Trigonometry
17.5
24. ANSWER: 22.2
26. ANSWER: 71.2
28. SNOWBOARDING Brad built a snowboarding ramp with a height of 3.5 feet and an 18º incline.
a. Draw a diagram to represent the situation.
b. Determine the length of the ramp.
ANSWER: a.
b. 11.3 ft
30. PARACHUTING A paratrooper encounters stronger winds than anticipated while parachuting from 1350 feet, causing him to drift at an 8º angle. How far from the drop zone will the paratrooper land?
ANSWER: 190 ft
Find the measure of angle θ. Round to the nearest degree, if necessary.
32. ANSWER: 48º
eSolutions Manual - Powered by Cognero
34. Page 3
32. ANSWER: 4-1 Right
Triangle Trigonometry
48º
34. ANSWER: 81º
36. ANSWER: 46º
38. ANSWER: 69º
40. OBSERVATION WHEEL The London Eye is a 135-meter-tall observation wheel. If a passenger at the top of
the wheel sights the London Aquarium at a 58º angle of depression, what is the distance between the aquarium and
the London Eye?
ANSWER: 84 m
42. SKI LIFT A company is installing a new ski lift on a 225-meter-high mountain that will ascend at a 48º angle of
elevation.
a. Draw a diagram to represent the situation.
b. Determine the length of cable the lift requires to extend from the base to the peak of the mountain.
ANSWER: a.
b. 303 m
eSolutions
Manual A tourist on the first observation level of the Eiffel Tower sights the Musée D’Orsay
- Powered by Cognero
44. PARIS
at a 1.4º angle ofPage 4
depression. A tourist on the third observation level, located 219 meters directly above the first, sights the Musée D’Orsay at a 6.8º angle of depression.
4-1 Right Triangle Trigonometry
b. 303 m
44. PARIS A tourist on the first observation level of the Eiffel Tower sights the Musée D’Orsay at a 1.4º angle of
depression. A tourist on the third observation level, located 219 meters directly above the first, sights the Musée D’Orsay at a 6.8º angle of depression.
a. Draw a diagram to represent the situation.
b. Determine the distance between the Eiffel Tower and the Musée D’Orsay.
ANSWER: a.
b. 2310 m
46. MOUNT RUSHMORE The faces of the presidents at Mount Rushmore are 60 feet tall. A visitor sees the top of George Washington’s head at a 48º angle of elevation and his chin at a 44.76º angle of elevation. Find the height
of Mount Rushmore.
ANSWER: about 500 ft
Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree.
48. ANSWER: X = 29º, y
37.1, z
32.5
50. ANSWER: D
77º, E
13º, d
29.2
52. eSolutions Manual - Powered by Cognero
ANSWER: W
14°, Y
76°, y
3.9
Page 5
50. ANSWER: 4-1 Right Triangle Trigonometry
D
77º, E
13º, d
29.2
76°, y
3.9
52. ANSWER: W
14°, Y
54. ANSWER: R
30º, S
60º, t
8.1
56. HIKING Jessica is standing 2 miles from the center of the base of Pikes Peak and looking at the summit of the
mountain, which is 1.4 miles from the base.
a. Draw a diagram to represent the situation.
b. With what angle of elevation is Jessica looking at the summit of the mountain?
ANSWER: a.
b.
Find the exact value of each expression without using a calculator.
58. cot 30°
ANSWER: 60. cos 45°
ANSWER: 62. csc 45°
ANSWER: Without using a calculator, find the measure of the acute angle θ in a right triangle that satisfies each
equation.
eSolutions Manual - Powered by Cognero
64. cos θ = Page 6
62. csc 45°
ANSWER: 4-1 Right Triangle Trigonometry
Without using a calculator, find the measure of the acute angle θ in a right triangle that satisfies each
equation.
64. cos θ = ANSWER: 30°
66. sin θ = ANSWER: 45°
68. sec θ = 2
ANSWER: 60°
Without using a calculator, determine the value of x.
70. ANSWER: 3
Find the value of cos θ if θ is the measure of the smallest angle in each type of right triangle.
72. 3-4-5
ANSWER: 0.8
74. SOLAR POWER Find the total area of the solar panel shown below.
ANSWER: 42.7 ft
2
Without using a calculator, insert the appropriate symbol >, <, or = to complete each equation.
76. tan 60° cot 30°
ANSWER: =
eSolutions Manual - Powered by Cognero
78. cos 30° sin 60°
ANSWER: Page 7
ANSWER: 4-1 Right2Triangle Trigonometry
42.7 ft
Without using a calculator, insert the appropriate symbol >, <, or = to complete each equation.
76. tan 60° cot 30°
ANSWER: =
78. cos 30° sin 60°
ANSWER: =
80. tan 45° sec 30°
ANSWER: < 82. MULTIPLE REPRESENTATIONS In this problem, you will investigate trigonometric functions of acute
angles and their relationship to points on the coordinate plane.
a. GRAPHICAL Let P(x, y) be a point in the first quadrant. Graph the line through point P and the origin. Form a
right triangle by connecting the points P, (x, 0), and the origin. Label the lengths of the legs of the triangle in terms
of x or y. Label the length of the hypotenuse as r and the angle the line makes with the x-axis θ.
b. ANALYTICAL Express the value of r in terms of x and y.
c. ANALYTICAL Express sin θ, cos θ, and tan θ in terms of x, y, and/or r.
d. VERBAL Under what condition can the coordinates of point P be expressed as (cos θ, sin θ)?
e. ANALYTICAL Which trigonometric ratio involving θ corresponds to the slope of the line?
f. ANALYTICAL Find an expression for the slope of the line perpendicular to the line in part a in terms of θ.
ANSWER: a.
b. r =
c. sin θ =
, cos θ =
, tan θ =
d. Sample answer: when r = 1
e . tan θ
f. –cot θ
84. ERROR ANALYSIS Jason and Nadina know the value of sin θ = a and are asked to find csc θ . Jason says that
this is not possible, but Nadina disagrees. Is either of them correct? Explain your reasoning.
ANSWER: Nadina; sample answer: The cosecant function is the reciprocal function of the sine function. Therefore, if sin θ =
eSolutions Manual - Powered by Cognero
a, then csc θ =
, where a ≠ 0.
Page 8
c. sin θ =
4-1
, cos θ =
, tan θ =
d. Sample answer: when r = 1
e . tan θTriangle Trigonometry
Right
f. –cot θ
84. ERROR ANALYSIS Jason and Nadina know the value of sin θ = a and are asked to find csc θ . Jason says that
this is not possible, but Nadina disagrees. Is either of them correct? Explain your reasoning.
ANSWER: Nadina; sample answer: The cosecant function is the reciprocal function of the sine function. Therefore, if sin θ =
a, then csc θ =
, where a ≠ 0.
86. CHALLENGE Write an expression in terms of θ for the area of the scalene triangle shown. Explain.
ANSWER: A=
; Sample answer: If you draw the height of the triangle, it forms two right triangles. The length of the
height is equal to a sin θ using the formula A =
bh, where the base of the triangle is b and the height is a sin θ.
REASONING If A and B are the acute angles of a right triangle and m A < m
each statement is true or false . If false, give a counterexample.
88. sin A < sin B
B, determine whether
ANSWER: true
90. tan A < tan B
ANSWER: true
92. ECONOMICS The Consumer Price Index (CPI) measures inflation. It is based on the average prices of goods and services in the United States, with the annual average for the years 1982-1984 set at an index of 100. The
table shown gives some annual average CPI values from 1955 to 2005. Find an exponential model relating this data
(year, CPI) by linearizing the data. Let x = 0 represent 1955. Then use your model to predict the CPI for 2025.
ANSWER: eSolutions Manual - Powered by Cognero
Sample answer: y = 24.2157e
0.0439x
; about 523.2
Page 9
90. tan A < tan B
ANSWER: 4-1 Right
Triangle Trigonometry
true
92. ECONOMICS The Consumer Price Index (CPI) measures inflation. It is based on the average prices of goods and services in the United States, with the annual average for the years 1982-1984 set at an index of 100. The
table shown gives some annual average CPI values from 1955 to 2005. Find an exponential model relating this data
(year, CPI) by linearizing the data. Let x = 0 represent 1955. Then use your model to predict the CPI for 2025.
ANSWER: Sample answer: y = 24.2157e
0.0439x
; about 523.2
Solve each equation. Round to the nearest hundredth.
94. 2ex – 7 – 6 = 0
ANSWER: 8.10
Sketch and analyze the graph of each function. Describe its domain, range, intercepts, asymptotes, end
behavior, and where the function is increasing or decreasing.
96. f (x) = 23x − 4 + 1
ANSWER: D = (–
,
), R = (1,
); no x-intercept, y-intercept:
increasing on (–
,
; horizontal asymptote at y = 1;
)
Solve each equation.
98. ANSWER: eSolutions
Manual - Powered by Cognero
–1, 8
Page 10
D = (–
,
), R = (1,
); no x-intercept, y-intercept:
increasing on (–
4-1 Right Triangle Trigonometry
,
; horizontal asymptote at y = 1;
)
Solve each equation.
98. ANSWER: –1, 8
100. ANSWER: 0, 1
102. SAT/ACT In the figure below, what is the value of z?
A 15
B 15
C 15
D 30
E 30
ANSWER: B
104. A person holds one end of a rope that runs through a pulley and has a weight attached to the other end. Assume
that the weight is at the same height as the person’s hand. What is the distance from the person's hand to the
weight?
A 7.8 ft
B 10.5 ft
C 12.9 ft
D 14.3 ft
ANSWER: A
eSolutions Manual - Powered by Cognero
Page 11