Fall Semester Final Exam Review
Name:
4-1 Review
Using the unit circle, find the following:
2𝜋
1. sin 3
2. cos 45°
3. tan 150°
4. sin
6. sec -330°
7. tan 2π
8. cos 𝜋
9. cot -90°
10. tan
12. cot 1800
13. sin –π
14. sec 45°
15. cos 4π
Change the degree measure to radian measure in terms of .
16. 20
17. -144
18. -300°
19. 18°
20. 160°
Change the radian measure to a degree.
𝜋
7𝜋
21. 4
22. − 6
24. 18
𝜋
11. csc − 6
23.
5𝜋
8
Change the following degrees to minutes and seconds.
26. 30.45°
27. 45.76°
𝜋
11𝜋
6
5. tan -45°
25. π
28. -102.37°
Convert the following degrees, seconds, and minutes back to a decimal. Round to 3 decimal places.
29. 102° 15’ 22”
30. 89° 44”
31. 92° 30’ 12”
4𝜋
3
Find a positive and negative coterminal angle for each angle.
32. 60°
33. -300°
34. 100°
35. Use the triangle shown to find all six trig ratios of the triangle.
Leave answers in simplest radical form.
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
3
5
Ɵ
36. In the previous problem, find the measure of θ and round to 2 decimal places.
1
37. If cosθ = 5, then sketch the triangle, find the missing side, and find all six trig ratios of the triangle.
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
Find the missing side lengths of the special right triangles below. Leave answers in simplest radical form.
38.
39.
40.
41.
42.
43.
44.
45.
Solve the right triangle. Round answers to 2 decimal places.
B
46.
C
47.
48.
B
C
C
A
A
A
B
C
49.
C
50.
51.
11
3
6
B
B
C
A
A
7
B
Solve. Round answers to 2 decimal places.
52. At 2pm, the shadow of a lighthouse is 19 feet long and the angle of elevation is 75. What is the height of the
lighthouse?
A
53. 13. A balloon is nailed to the ground by its 70 foot string. The balloon is blowing in the breeze and making a 55o angle
with the ground. How high is the balloon in the air?
54. Calvin creates a zip line from a tree to the ground. The line creates a 63o angle with the tree. The tree is 25 feet tall.
How long is the zip line?
If Calvin can get down the zip line in 6 seconds, how fast is he traveling?
55. A biologist wants to know the width, w, of a river. From point A, the biologist walks downstream 93feet to point B
and sights point C (see figure to the right). From this sighting, it is determined that θ = 46°. How wide is the river?
56. A line that is 45 meters long is used to tether a helium-filled balloon. Because of a breeze, the line makes an angle of
approximately 65° with the ground. What is the vertical height of the balloon? Round your answer to two decimal
places and sketch a diagram.
4-2 Review
State the amplitude, period, phase shift, and vertical shift.
57. y  3sin 2 x
1
2
58. y  2  cos( x   )
59. y  sin(4 x 

2
)3
60. y  1.25cos(

4
x  2 )
1
3
61. y  sin x  2
62. y  1  10 cos( x 

6
Use the given information to write the equation of the sine function.
63. Amp: 1
Period: 2π
PS:

2
1
2
Period: -π
PS:

3
65. Amp:
VS: up 4

2
64. Amp: 2
Period:
66. Amp: 10
Period: 8π
PS:

4
PS: 
Graph the following trig functions. (Label the positive tick marks)
67. y = 2 sin (2x)
1
2
68. y = -3 cos (x)
69. y = 1 + cos ( x)
70. y = 2 sin (4x + 2π)
71. y = 4 csc (x – π)
72. y = 3 - sec (4x)
VS: down 1
VS: down 1
)
1
74. 3csc ( 4x )
73. y = sec (3x)
75. y = 2 tan (3x)
76. y = cot (5x)
77. y = - cot (7x)
4-3 Review
Solve the following trig equations. Give answers within the interval of 0 to 2π.
78. 2sin x  1  0
79. cot x 1  0
80. 2 cos 2 x  1  0
81.
3 tan x  1  0
82.
1
sec x  1
2
83. csc 2 x  2  0
84. sin 2 x  sin x
85. 2cos x  cos x  3
86. sin x  cos x
87. cot 2 x  3
88. sin x  3  sin x
89. 3sec 2 x  4
Arc Length, Angular Velocity, and Linear Velocity
Use the following formulas to answer the questions below.
S = rθ
𝝎=
𝜽
𝒕
ν=r𝝎
90. A flywheel rotates with an angular velocity of 2 rps. Find the linear velocity in inches per minute if the radius is 15 in.
91. The radius of a soccer ball is 18 cm. What is the length of an arc of the ball for a central angle of 45o.
92. Find the length of a pendulum if it oscillates through an angle of 10°, and swings a distance of 6 in. from one end
to the other.
93. Calculate the linear velocity of a reflector located 10 in. from the center of a bicycle wheel rotating 7 rad per sec.
94. A compact disc is spinning with an angular speed of 4.3 rotations per second.
What is its angular speed in radians per minute?
3
2
Key: 1)
15) 1 16)
2)

9
2
2
17)
3)

4
5

3
3
18)
4)


1
2
5
3
5) -1 6)
19)

2 3
3
20)
10
8
9
7) 0 8) -1 9) 0 10) 3 11) -2
12) UND 13) 0 14) 2
21) 45° 22) -210° 23) 112.5° 24) 10° 25. 180° 26) 30° 27’
27) 45° 45’ 36” 28) -102° 22’ 12” 29) 102.256 30) 89.012 31) 92.503 32) 420°, -300° 33) 60°, -660° 34) 460°, 260°
35)
3
4
3
5
5
4
sin   ,cos  , tan   ,csc  ,sec  ,cot  
5
5
4
3
4
3
37)
sin  
36) 36.87°
2 6
1
5 6
6
,cos  , tan   2 6,csc 
,sec  5,cot  
5
5
12
12
41) s  3 3, t  6 3 42) a  4, b  2 2
38) x  5 3, y  10 39) x  5 2, y  5 2 40) x  24, y  12 3
43) x =10; y = 5 44) x  4 15, y  4 5 45) x  2 3, y  2 3 46) AB= 7.28, BC= 5.29,
<C= 54°
47) AB= 27.29, AC=88.32, <C=18° 48) AB= 14.85, BC= 14.85, <C=45° 49) AB= 9.22 <A= 33.06° <C= 56.94° 50) AC= 7.62
<A= 23.2° <C= 66.8°
51) BC= 5.6, AC= 13.24, <A= 25° 52) 70.91 ft 53) 57.34 ft 54) 55.07 ft , 9.18sec 55) w= 96.3 ft 56) h = 40.78 ft 57)
Amp= 3, Per= π


58) Amp= 1, Per= 4π, PS= -2π, VS= up 2 59) Amp= 1, Per= 2 PS= 8 VS: down 3 60) Amp= 1.25, Per=8, PS= -8 VS=
none


61) Amp=1, Per= 2π, PS= none, VS= down 2 62) Amp=10, Per= 6π, PS= 2 , VS= down 1 63) y  sin( x  2 ) 64)
y  1  2sin(4 x   )
1
2
1

65) y  2 sin(2 x  3 )  4 66) y  1  10sin( 4 x  4 ) 67) – 77) graphs
7 11
78) 6 , 6
 5
79) 4 , 4
 3 5 7
80) 4 , 4 , 4 , 4 81)
5 11
,
6 6
 5
82) 3 , 3
 3 5 7
83) 4 , 4 , 4 , 4
90) 1800 inch/min

 5
84) 2 ,  , 2 85) π 86) 4 , 4
91) 14.14 cm
 5 7 11
87) 6 , 6 , 6 , 6
92) 108 inches 93) 219.9 inches
 2
88) 3 , 3
 5 7 11
89) 6 , 6 , 6 , 6
94) 1621.06 rad/min