5.2e: Exercises - Right Angle Trigonometry
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A: Given three sides of a right triangle, find all six trigonometric ratios
Exercise 5.2e.A
★ Given right triangle ABC where the right angle is angle C in each figure below,
(a) Label the remaining sides and angles
(b) Designate the hypotenuse, adjacent side or opposite side to angle A.
Determine the trigonometric ratios for (c) sinA, (d) cosA, (e) tanA, (f) secA, (g) cscA, (h) cotA.
Give simplified exact answers - reduce fractions, rationalize all denominators!
- Answers to odd exercises.
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1. (c) sinA=45, (d) cosA=35, (e) tanA=43,
(f) secA=53, (g) cscA=54, (h) cotA=343. (c) sinA=√53, (d) cosA=23, (e) tanA=√52,
(f) secA=32, (g) cscA=3√55, (h) cotA=2√55
B: Given two sides of right triangle, find all trigonometric ratios of the acute angles
Exercise 5.2e.B
★ Find the exact values of all six trigonometric functions of angles A and B in the right triangle △ABC illustrated below. Simplify!
5. b=12, c=13 6. b=15, c=17 7. a=7, c=25 |
8. a=20, c=29 9. a=9, b=40 10. a=1, b=2 |
11. a=1, b=3 12. a=2, b=5 13. a=5, c=6 |
14. a=2, c=√6 15. b=7, c=8 16. b=3, c=√15 |
★ In each of the triangles below, find sin(A),cos(A),tan(A),sec(A),csc(A),cot(A).
17. 18.
- Answers to odd exercises.
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For all triangles #5-16, the trigonometric ratios can be determined using the following formulas:
sinA=cosB=ac, cosA=sinB=bc, tanA=cotB=ab,
cscA=secB=ca, secA=cscB=cb, cotA=tanB=ba,
The missing side needed to calculate these ratios is given below:
5. a=5 7. b=24 9. c=41 11. c=√10 13. b=√11 15. a = √1517. sin(A)=5√4141,cos(A)=4√4141,tan(A)=54,sec(A)=√414,csc(A)=√415,cot(A)=45
C: Given one trigonometric ratio of an acute angle, find all the others
Exercise 5.2e.C
★ Find the exact values of the other five trigonometric functions of the acute angle A given the indicated value of one of the functions. Simplify!
21. sinA=12 22. sinA=512 23. cosA=15 |
24. cosA=23 25. tanA=34 26. tanA=95 |
27. cscA=73 28. cscA=√31 29. secA=3 |
30. secA=5√63 31. cotA=10 32. cotA=4 |
- Answers to odd exercises.
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21. sinA=12, cosA=√32, tanA=√33, cscA=2, secA=2√33, cotA=√3
23. sinA=2√65, cosA=15, tanA=2√6, cscA=5√612, secA=5, cotA=√612
25. sinA=35, cosA=45, tanA=34, cscA=53, secA=54, cotA=43
27. sinA=37, cosA=2√107, tanA=3√1020, cscA=73, secA=7√1020, cotA=2√103
29. sinA=2√23, cosA=13, tanA=2√2, cscA=3√24, secA=3, cotθ=√24
31. sinA=√101101, cosA=10√101101, tanA=110, cscA=√101, secA=√10110, cotA=10
D: Cofunctions
Exercise 5.2e.D
Use cofunctions of complementary angles to write an equivalent expression.
36. cos(34°)=sin(__°) | 37. \cos (\dfrac{π}{3})= \sin (\_\_\_) | 38. \csc (21°) = \sec (\_\_\_°) | 39. \tan (\dfrac{π}{4})= \cot (\_\_) |
- Answers to odd exercises.
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37. \dfrac{π}{6} 39. \dfrac{π}{4}
E: Given one side and an acute angle of a right triangle, find the other sides and angles
Exercise \PageIndex{E}
\bigstar In each of the following triangles, solve for x to the nearest tenth.
41. |
42. |
43. |
44. |
45. |
46. |
47. |
48. |
49. |
50. |
51. |
52. |
- Answers to odd exercises.
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41. 6.4 43. 7.7 45. 11.9 47. 8.4 49. 44.8 51. 7.8
\bigstar In each of the following triangles, solve for x to the nearest tenth.
53. |
54. |
55. |
56. |
57. |
58. |
59.
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60, |
- Answers to odd exercises.
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53. 20.5 55. 14.5 57. 7.3 59. 4.8
\bigstar In each of the following triangles, solve for the unknown sides and angles. Give answers to 4 decimal digits.
61. 62.
63.
64. 65.
66.
67. 68.
69. a=5, angle opposite side a is ∡ A=60^∘
70. Hypotenuse c=12, and one acute angle is ∡ A=45^∘
\bigstar Find x. Give answers to 4 decimal digits.
71. 72.
73. 74.
- Answers to odd exercises.
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61. b \approx 12.1244, c =14, B = 60^{\circ} 63. a \approx 5.3171, c \approx 11.3257, A = 28^{\circ} 65. a \approx 9.0631, b \approx 4.2262, B = 25^{\circ}
67. a = 15, b = 15, A = 45^{\circ} 69. b \approx 2.8868, c \approx 5.7735, B = 30^{\circ} 71. 188.3159 73. 200.6737
F: Right Triangle Applications
Exercise \PageIndex{F}
Solve. Round answers to the nearest 10^{th} unless otherwise specified.
81. A 23-ft ladder leans against a building so that the angle between the ground and the ladder is 80°. How high does the ladder reach up the side of the building?
82. A 33-ft ladder leans against a building so that the angle between the ground and the ladder is 80°. How high does the ladder reach up the side of the building?
83. A 20 foot ladder is leaning against a wall, It makes an angle of 70^{\circ} with the ground. How high is the top of the ladder from the ground (nearest tenth of a foot)?
84. A 275 foot guy wire is attached to the top of a communication tower. If the wire makes an angle of 53^{\circ} with the ground, how tall is the tower?
ANGLE OF ELEVATION

85. At a point 50 feet from a tree the angle of elevation of the top of the tree is 43^{\circ}. Find the height of the tree to the nearest tenth of a foot.
86. At a point 60 feet from a tree the angle of elevation of the top of the tree is 40^{\circ}. Find the height of the tree to the nearest tenth of a foot.
87. The angle of elevation to the top of a mountain from a point 20 miles away from the base of the mountain is 6^{\circ}. How high is the mountain to the nearest foot? One mile is 5280 feet.

89. At a point 100 feet from a tall building the angle of elevation of the top of the building is 65^{\circ}. Find the height of the building to the nearest foot.
90. The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building in feet. One mile is 5280 feet.
91. The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building in feet. One mile is 5280 feet.
93. Assuming that a 370-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be 60°, how far from the base of the tree am I?
94. The CN Tower in Toronto is 1815 feet tall. When the angle of elevation to the top of the tower is observed to be 40°, how far away from the Tower is that location?
ANGLE OF DEPRESSION


95. From the top of a building 125 feet tall, the angle of depression of an intersection is 34^{\circ} . How far from the base of the building is the intersection?
Notice that the angle of elevation will be equal to the corresponding angle of depression.
96. A helicopter that is 700 feet in the air measures the angle of depression to a landing pad as 24^{\circ} . How far is the landing pad from the point directly beneath the helicopter's current position?

97. From the top of a 100 foot lighthouse the angle of depression of a boat is 15^{\circ}. How far is the boat from the bottom of the lighthouse (nearest foot)?
98. From the top of a lighthouse 180 feet above sea level, the angle of depression to a ship in the ocean is 28^{\circ} . How far is the ship from the base of the lighthouse?

99. From an airplane 5000 feet above the ground the angle of depression of an airport is 5^{\circ}. How far away is the airport to the nearest hundred feet?
100. From a helicopter 1000 feet above the ground the angle of depression of a heliport is 10^{\circ}. How far away is the heliport to the nearest foot?
TWO ANGLES of DEPRESSION and/or ELEVATION
101. A radio tower is located 325 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 43°, and that the angle of depression to the bottom of the tower is 31°. How tall is the tower?
102. A radio tower is located 400 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 36°, and that the angle of depression to the bottom of the tower is 23°. How tall is the tower?
103. A 400-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is 18°, and that the angle of depression to the bottom of the monument is 3°. How far is the person from the monument?
104. A 200-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is 15°, and that the angle of depression to the bottom of the tower is 2°. How far is the person from the monument?
105. There is lightning rod on the top of a building. From a location 500 feet from the base of the building, the angle of elevation to the top of the building is measured to be 36°. From the same location, the angle of elevation to the top of the lightning rod is measured to be 38°. Assuming the lightning rod is situated at the edge of the building facing the observer, find the height of the lightning rod.
106. There is an antenna on the top of a building. From a location 300 feet from the base of the building, the angle of elevation to the top of the building is measured to be 40°. From the same location, the angle of elevation to the top of the antenna is measured to be 43°. Find the height of the antenna.
- Answers to odd exercises.
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81. 22.7 ft 83. 18.8 ft 85. 46.6 feet. 87. 11,099 feet. 89. 214 feet. 91. 368.8 ft 93. 213.6 ft
95. d \approx 185.32 ft 97. 373.2 ft 99. 57,300 feet 101. 498.3 ft 103. 1060.1 ft 105. 27.4 ft
G: Find an angle given 2 sides of a right triangle
Exercise \PageIndex{G}
\bigstar Find the angle x to the nearest tenth of a degree:
111. |
112. |
113. |
114. |
115. |
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117. |
118. |
- Answers to odd exercises.
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111. 41.8^{\circ} 113. 36.9^{\circ} 115. 56.3^{\circ} 117. 48.2^{\circ}
H: Applications Finding an Angle
Exercise \PageIndex{H}
121. If a 20 foot telephone pole casts a shadow of 43 feet , what is the angle of elevation of the sun to the nearest tenth?
122. An 88 foot tree casts a shadow that is 135 feet long. What is the angle of elevation of the sun to the nearest tenth?
123. A road rises 30 feet in a horizontal distance of 300 feet. Find the angle the road makes with the horizontal to the nearest tenth of a degree.
124. A road rises 10 feet in a horizontal distance of 400 feet. Find the angle the road makes with the horizontal to the nearest tenth of a degree.
- Answers to odd exercises.
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121. 24.9^{\circ} 123. 5.7^{\circ}
I: Calculator Practice
Exercise \PageIndex{I}
\bigstar Use a calculator to find the value to four decimal digits. Be sure your calculator is in the appropriate mode!
131. \sin 10^{\circ} 132. \sin 30^{\circ} 133. \cos 80^{\circ} 134. \cos 60^{\circ} 135. \tan 45^{\circ}
136. \tan 60^{\circ} 137. \sin 18^{\circ} 138. \cos 72^{\circ} 139. \tan 50^{\circ} 140. \tan 80^{\circ}
141. \sin \left( \dfrac{\pi}{12} \right) 142. \sin \left( \dfrac{5\pi}{12} \right) 143. \cos \left( \dfrac{3\pi}{8} \right) 144. \cos \left( \dfrac{3\pi}{8} \right)
145. \tan \left( \dfrac{\pi}{3} \right) 146. \tan \left( 0.8 \right) 147. \sin \left( 1.5 \right) 148. \cos \left( 1.0 \right)
- Answers to odd exercises.
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131. 0.1736 133. 0.1736 135. 1 137. 0.3090 139. 1.1918
141. 0.2588 143. 0.3827 145. 1.7321 147. 0.9975
\bigstar Use a calculator to find the value (a) in degrees and (b) in radians to four decimal digits.
151. \sin^{-1} (.1) 152. \sin^{-1} (.1) 153. \cos^{-1} (.3) 154. \cos^{-1} (.9)
155. \tan^{-1} (.5) 156. \tan^{-1} (4) 157. \sin^{-1} (3) 158. \cos^{-1} (2)
- Answers to odd exercises.
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151. (a) 6.7392^{\circ} (b) .1002 (radians) 153. (a) 72.5425^{\circ} (b) 1.2661 (radians)
155. (a) 26.5651^{\circ} (b) 1.3258 (radians) 157. Undefined ("Domain Error")