Download Trigonometry of the Right Triangle – Class Work Unless otherwise

Trigonometry of the Right Triangle – Class Work Unless otherwise directed, leave answers
as reduced fractions or round to the nearest tenth.
1. Evaluate the sin, cos, and tan of 𝜃(𝑡ℎ𝑒𝑡𝑎).
2. Evaluate the sin, cos, and tan of 𝛼 (𝑎𝑙𝑝ℎ𝑎).
3. Find x. Evaluate sin, cos, and tangent of 𝛽 (𝑏𝑒𝑡𝑎) and 𝛾 (𝑔𝑎𝑚𝑚𝑎).
Round answers to the nearest hundredth.
4. Find x.
9
5. A right triangle has an acute angle of 𝛼, and cos 𝛼 = 41. What is sin 𝛼?
6. A right triangle has a hypotenuse of 7 and an angle of 40°, find the larger leg (in any
triangle, the longest side is opposite the largest angle, the shortest side is opposite the
smallest angle, and the middle side is opposite the middle angle).
7. A right triangle has an angle of 50° and a longer leg of 8, find the hypotenuse.
8. Solve right triangle ABC using the measurements provided and the diagram shown.
a. B = 50°, a = 10
c. A = 36°, c = 20
b. A = 72°, a = 12
d. B = 17°, b = 24
9. Evaluate all six trig functions of the angle, 𝛽. Give answers as exact values.
a.
b.
10. Let 𝜃 be an acute angle in a right triangle. Find the values of the other 5 trig functions.
Give answers as exact values in simplified form.
a. sin 𝜃 =
Alg II – Triangle Trig
8
9
b. cos 𝜃 =
√3
4
~1~
c. csc 𝜃 =
10
3
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Trigonometry of the Right Triangle – Homework Round answers to the nearest tenth.
11. Evaluate the sin, cos, and tan of 𝜃(𝑡ℎ𝑒𝑡𝑎).
12. Evaluate the sin, cos, and tan of 𝛼 (𝑎𝑙𝑝ℎ𝑎).
13. Find x. Evaluate sin, cos, and tangent of 𝛽 (𝑏𝑒𝑡𝑎) and 𝛾 (𝑔𝑎𝑚𝑚𝑎).
Round answers to the nearest hundredth.
14. Find x. Evaluate tangent of 17°.
9
15. A right triangle has an acute angle of 𝛼, and sin 𝛼 = 41. What is tan 𝛼?
16. A right triangle has a hypotenuse of 9 and an angle of 60°, find the larger leg.
17. A right triangle has an angle of 20° and a longer leg of 5, find the hypotenuse.
18. Solve right triangle ABC using the measurements provided and the diagram shown.
a. B = 40°, a = 19
c. A = 28°, c = 11
b. A = 50°, a = 3.1
d. B = 35°, b = 24
19. Evaluate all six trig functions of the angle, 𝛽. Give answers as exact values in simplified
form.
a.
b.
20. Let 𝜃 be an acute angle in a right triangle. Find the values of the other 5 trig functions.
Give answers as exact values in simplified form.
a. sin 𝜃 =
Alg II – Triangle Trig
3
8
b. csc 𝜃 =
3√2
4
~2~
c. tan 𝜃 =
10
3
NJCTL.org
Inverse Trig Functions – Class Work
20. 𝜃 =
22. 𝛼 =
23. 𝛽 =
24. 𝛾 =
25. A right triangle has legs of 7 and 12. Find the smaller acute angle.
26. A right triangle has a longer leg of 5, and a hypotenuse of 13. Find the larger acute
angle.
Inverse Trig Functions – Homework
27. 𝜃 =
28. 𝛼 =
29. 𝛽 =
30. 𝛾 =
31. A right triangle has legs of 6 and 10. Find the larger acute angle.
32. A right triangle has a longer leg of 1.2, and a hypotenuse of 2. Find the larger acute
angle.
Problem-Solving Classwork
33. A tree is 75 feet tall. What is the length of its shadow if the angle it makes with the sun is
55°?
34. Sally is wearing 10-inch heels. IF the platform of her shoe is 5 inches,
and her foot from heel to arch is 8 inches, what is the angle of elevation
of her foot?
35. Tony spots a radio tower at the top of a building. He is standing 50 meters from the base
of the building. If the angles of elevation of his sight are 58° and 62° to the bottom and the
top of the tower respectively, what is the height of the tower?
Problem-Solving Homework
36. Snow White is being held prisoner in a tower with a window 40 feet from the ground.
Prince Charming shows up with a 45-foot ladder. If the maximum angle that the ladder
can make with the ground is 72°, will the prince be able to rescue Snow White? Expain
your answer.
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37. A ski lift takes skiers 500 meters up from the base of a ski trail to the top. If the elevation
at the top is 2200 meters and the elevation at the bottom is 1800 meters, what is the
angle of elevation of the ski lift?
38. Two points in front of a building are 25 feet apart. The angles of elevation from the points
to the top of the building are 18° and 25°. How tall is the building?
Special Right Triangles Classwork
39. g =
40. m =
h=
n=
41. p =
q=
Special Right Triangles Homework
42. g =
43. m =
h=
n=
44. p =
q=
Law of Sines – Class Work
Solve triangle ABC.
45. 𝐴 = 70°, 𝐵 = 30°, 𝑐 = 4
.
46. 𝐵 = 65°, 𝐶 = 50°, 𝑎 = 12
47. 𝑏 = 6, 𝐴 = 25°, 𝐵 = 45°
48. 𝑐 = 8, 𝐵 = 60°, 𝐶 = 40°
49. 𝑐 = 12, 𝑏 = 6, 𝐶 = 70°
50. 𝑏 = 12, 𝑎 = 15, 𝐵 = 40°
51. 𝐴 = 35°, 𝑎 = 6, 𝑏 = 11
52. A swimmer is swimming in the ocean and gets a leg cramp in between two lifeguard
stands. If the stands are 320 feet apart, and the angles from the stands to the swimmer
are 50° and 45°, find the distance from each lifeguard to the swimmer.
53. You are creating a triangular patio. One side is 29 feet long and another is 32 feet. The
angle opposite the 32 ft. side is 75°. If you want to put a brick border around the patio,
how many feet of border do you need? If each brick is 8 inches long, about how many
bricks will you need?
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Law of Sines – Homework
Solve triangle ABC.
54. 𝐴 = 60°, 𝐵 = 40°, 𝑐 = 5
55. 𝐵 = 75°, 𝐶 = 50°, 𝑎 = 14
56. 𝑏 = 6, 𝐴 = 35°, 𝐵 = 45°
57. 𝑐 = 8, 𝐵 = 50°, 𝐶 = 40°
58. 𝑐 = 12, 𝑏 = 8, 𝐶 = 65°
59. 𝑏 = 12, 𝑎 = 16, 𝐵 = 50°
60. 𝐴 = 40°, 𝑎 = 5, 𝑏 = 12
61. A triangular pane in a stained glass window has two angles of 35° and 85°. If the side
between these angles is 75 cm, what are the lengths of the other two sides?
62. A roller coaster has a hill that goes up at a 70° angle with the ground. It gets to the top
and turns 30° before dropping 418 feet. What is the length of the upward climb?
Law of Cosines – Class Work
Solve triangle ABC.
63. 𝑎 = 3, 𝑏 = 4, 𝑐 = 6
64. 𝑎 = 5, 𝑏 = 6, 𝑐 = 7
65. 𝑎 = 7, 𝑏 = 6, 𝑐 = 4
66. 𝐴 = 100°, 𝑏 = 4, 𝑐 = 5
67. 𝐵 = 60°, 𝑎 = 5, 𝑐 = 9
68. 𝐶 = 40°, 𝑎 = 10, 𝑏 = 12
69. The airline distance from Los Angeles, CA to Phoenix, AZ is 367 miles. The distance
from Phoenix to Salt Lake City, UT is 504 miles and the distance from Salt Lake City to
Los Angeles is 579 miles. What is the angle formed from LA to Phoenix to Salt Lake
City?
70. Cal C takes his 2 dogs for a walk. He lets them off their leash in a field where Edison runs
at 7 m/s and Einstein runs at 6 m/s. Cal determines the angle between the dogs is 20°,
how far are the dogs from each other in 8 seconds?
71. A golfer hits a ball 30° to the left of straight towards the hole. If the length of his shot is
156 yards, and the length from the tee to the hole is 186 yards, how far is the golfer’s ball
from the tee?
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Law of Cosines – Home Work
Solve triangle ABC.
72. 𝑎 = 4, 𝑏 = 5, 𝑐 = 8
73. 𝑎 = 4, 𝑏 = 10, 𝑐 = 13
74. 𝑎 = 11, 𝑏 = 8, 𝑐 = 6
75. 𝐴 = 85°, 𝑏 = 3, 𝑐 = 7
76. 𝐵 = 70°, 𝑎 = 6, 𝑐 = 12
77. 𝐶 = 25°, 𝑎 = 14, 𝑏 = 19
78. The airline distance from Rome, Italy to Paris, France is 697 miles. The distance from
Paris to Berlin, Germany is 545 miles and the distance from Berlin to Rome is 734 miles.
What is the angle formed from Berlin to Paris to Rome?
79. A student takes his 2 dogs for a walk. He lets them off their leash in a field where Edison
runs at 10 m/s and Einstein runs at 8 m/s. The student determines the angle between the
dogs is 25°, how far are the dogs from each other in 5 seconds?
80. A golfer hits a ball 15° to the right of straight towards the hole. If the length of his shot is
89 yards, and the length from the tee to the hole is 160 yards, how far is the golfer’s ball
from the tee?
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Unit Review
Multiple Choice
1. Evaluate tan 𝛼.
a. 0.9
b. 0.45
c. 0.5
d. 2
2. The longer leg of a right triangle is 6 and the smallest angle is 20°, what is the hypotenuse?
a. 6.4
b. 5.6
c. 17.5
d. 14.7
3. Find the value of a.
a. 18.8
b. 0.03
c. 15
d. 20
4. Find the value of 𝛽.
a. 56.3°
b. 41.8°
c. 33.7°
d. 48.2°
5. Find the exact value of sin 𝛽 in simplest form.
a. 18.0
b.
c.
d.
2√13
13
10
√325
3√325
65
6. Given ∆𝐴𝐵𝐶, 𝑤𝑖𝑡ℎ 𝐴 = 35°, 𝑎 = 5, & 𝑐 = 7, 𝑓𝑖𝑛𝑑 𝐵.
a. 18.418
b. 53.418
c. 91.582
d. both a and b
7
7. Let 𝜃 be an acute angle in a right triangle. If sec 𝜃 = , what is sin 𝜃?
6
a.
b.
c.
d.
1
7
√13
6
7
√13
√13
7
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8. Given ∆𝐴𝐵𝐶, 𝑤𝑖𝑡ℎ 𝐴 = 50°, 𝑎 = 6, & 𝑐 = 8, 𝑓𝑖𝑛𝑑 𝐵.
a. 1.021
b. 40
9. Given ∆𝐴𝐵𝐶, 𝑤𝑖𝑡ℎ 𝑎 = 9, 𝑏 = 6, & 𝑐 = 8, 𝑓𝑖𝑛𝑑 𝐵.
a. 6.188
b. 42.6
c. 40.8
d. 78.6
c. 128.979
d. no solution
Extended Response
1. A state park hires a surveyor to map out the park.
a. A and B are on opposite sides of the lake, if the surveyor stands at point C and
measures angle ACB= 50 and CA= 400’ and CB= 350’, how wide is the lake?
b. At a river the surveyor picks to spots, X and Y, on the same bank of the river and a
tree, C, on opposite bank. X= 80 and Y= 50 and XY=300’, how wide is the river?
(Remember distance is measured along perpendiculars.)
c. The surveyor measured the angle to the top of a hill at the center of the park to be
32°. She moved 200’ closer and the angle to the top of the hill was 43°. How tall
was the hill?
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2. A 15’ ladder is rated to have no more than a 70° angle and no less than a 40° angle.
a. What is maximum rated distance the base of the ladder can be placed from the
wall?
b. How high up a wall can the ladder reach and be within the acceptable use limits?
c. At what base angle should the ladder be placed to reach 10’ up the wall?
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Answer Key
1. sin 𝜃 = 8⁄17 , cos 𝜃 = 15⁄17 , tan 𝜃 = 8⁄15
2. sin 𝛼 = 4⁄5 , cos 𝛼 = 3⁄5 , tan 𝛼 = 4⁄3
3. x = 14.4, sin 𝛽 = . 83, cos 𝛽 =
.56, tan 𝛽 = . 67
13. x ≈ 9.7, sin 𝛽 = . 58, cos 𝛽 =
.81, tan 𝛽 = . 72
sin 𝛾 = . 81, cos 𝛾 = .58, tan 𝛾 = 1.39
14. x = 7.7, cos 17° = 0.96
sin 𝛾 = . 56, cos 𝛾 = .83, tan 𝛾 = 1.5
15.
4. x = 14.3
5.
16. 7.8
40
41
17. 5.3
6. 5.4
18. a.
b.
c.
d.
7. 10.4
8. a.
b.
c.
d.
9
40
b = 11.9 , c = 15.6 , A = 40°, C = 90°
b = 3.9, c = 12.6, B = 18, C = 90°
a = 11.8, b = 16.2, B = 54°, C = 90°
a = 78.5, c = 82.1, A = 73°, C = 90°
3
5
4
5
9. a. Sin 𝛽 = , cos 𝛽 = , tan 𝛽 =
5
4
b. Sin 𝛽 =
csc 𝛽
b. Sin 𝛽 = .7, cos 𝛽 = .7, tan 𝛽 = 1
10. a. cos 𝜃 =
10
, sec 𝛽
7
=
csc 𝜃 =
cot 𝛽 = 1
√17
, tan 𝜃
9
=
9
8
9√17
,
17
csc 𝜃 = , sec 𝜃 =
b. Sin 𝜃 =
10
,
7
√13
4
4√13
, sec 𝜃
13
3
c. Sin 𝜃 = 10 , cos 𝜃 =
sec 𝜃 =
=
cot 𝜃 =
√17
8
b. Sin 𝜃 =
√39
3
cot 𝜃 =
=
√39
13
c. Sin 𝜃 =
3√91
91
=
=
8
8√55
,
55
√21
2
cot 𝜃 =
√55
3
, tan 𝜃 = 2√2
√2
4
10√109
, cos 𝜃
109
csc 𝜃 =
cot 𝛽 =
2√21
21
3√55
55
=
2√2
1
, cos 3
3
=
4√137
11
, tan 𝛽 = 4
137
4
√137
, cot 𝛽 = 11
4
√55
, tan 𝜃
8
sec θ = 3, cot 𝜃 =
√91
, tan 𝜃
10
=
5√21
,
21
csc 𝜃 = 3 , sec 𝜃 =
4√3
,
3
10√91
, cot 𝜃
91
5
2
11√137
, cos 𝛽
137
√137
= 11 , sec 𝛽
20. a. cos 𝜃 =
8√17
17
, tan 𝜃 =
√21
, tan 𝛽
5
csc 𝛽 = , sec 𝛽 =
csc 𝛽 = 3 , sec 𝛽 = 4, cot 𝛽 = 3
csc 𝛽 =
2
5
19. a. Sin 𝛽 = , cos 𝛽 =
3
4
5
b = 15.9 , c = 24.8 , A = 50°, C = 90°
b = 2.6, c = 4, B = 40°, C = 90°
a = 5.2, b = 9.7, B = 62°, C = 90°
a = 34.3, c = 41.8, A = 55°, C = 90°
√109
sec 𝜃
10
=
=
3√109
,
109
√109
, cot 𝜃
3
3
= 10
21. 28.1°
√91
3
22. 53.1°
11. sin 𝜃 = 5⁄13 , cos 𝜃 = 12⁄13 , tan 𝜃 = 5⁄12
23. 56.3°
12. sin 𝛼 = 117⁄125 , cos 𝛼 = 44⁄125 ,
tan 𝛼 = 117⁄44
24. 33.7°
25. 30.3°
Alg II – Triangle Trig
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26. 22.6°
53. 84 feet, about 126 bricks
27. 22.6°
54. C = 80°, a = 4.40, b = 3.26
55. A = 55°, b = 16.51, c = 13.09
28. 69.4°
56. C = 100°, a = 4.87, c = 8.36
29. 35.7°
57. A = 90°, a = 12.45, c = 9.53
30. 54.3°
58. A = 77.83°, B = 37.17°, a = 12.94
31. 59°
59. No Solution
32. 36.9°
60. No Solution
33. 107.1 ft
61. 438.1 ft
34. 32°
62. 49.7 cm and 86.3 cm
35. 14 m
63. A = 26.384°, B = 36.336°, C = 117.280°
36. Yes, the ladder can reach a maximum height
64. A = 44.415°, B = 57.122°, C = 78.463°
of 43 feet.
65. A = 86.417°, B = 58.811°, C = 34.772°
37. 53.1°
66. a = 6.924, B = 34.670°, C = 45.330°
38. 27 feet (hint: set up and solve a system of
67. A = 33.668°, b = 7.810, C = 86.332°
equations)
68. A = 55.978°, B = 84.022°, c = 7.756
39. g = 6, h = 6√3
69. 81.7°
40. 𝑚 = 4.5√3, n = 4.5
70. 19.7 m
41. p = 5√2, 𝑞 = 5√2
71. 93.1 yd
2
2√3
3
42. 𝑔 = 3 , ℎ =
43. m =
16√3
,𝑛
3
=
72. A = 24.147°, B = 30.754°, C = 125.1°
73. A = 13.325°, B = 35.184°, C = 131.491°
8√3
3
74. A = 102.636°, B = 45.207°, C = 32.157°
44. p = 15, q = 15√2
75. a = 7.372, B = 23.916°, C = 71.074°
45. C = 80°, a = 3.82, b = 2.03
76. b = 11.435, A = 29.543°, C = 80.453°
46. A = 65°, b = 12, c = 10.14
77. c = 8.651, A = 43.149°, B = 111.852°
47. C = 110°, a = 3.59, c = 7.97
78. 71.3°
48. A = 80°, a =12.26, b = 10.78
79. 21.8 m
49. A = 82°, B = 28°, a = 12.65
80. 77.5 yd
50. A = 53.5°, C = 86.5°, c = 18.63
MC1. C
and A = 126.5°, C = 13.5°, c = 4.35
MC2. A
51. No Solution
MC3. A
52. 227.1 feet, 246.1 feet
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MC4. B
MC5. B
MC6. C
MC7. D
MC8. D
MC9. C
ER1A. 320.2’
ER1B. about 295’ (hint: use 2 right triangles, set
up and solve a system of 2 equations)
ER1C. about 385’
ER2A. 11.5’
ER2B. 14.1’
ER2C. 41.8°
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