Pre-Calculus – Trigonometric Functions ~1~ NJCTL.org
PRECALCULUS UNIT 4: TRIGONOMETRY PRACTICE PROBLEMS Angle and Radian Measures Convert each degree measure into radians. Round answers to the 4th decimal place.
1. 34.375°
2. 176.48°
3. 225.8525°
Convert each radian measure into degrees. Round answers to the 4th decimal place.
4. 0.25 radians
5. 1.34 radians
6. 4.28 radians
Find the length of each arc.
7. 𝜃 = 235°, 𝑟 = 9 𝑐𝑚
8. 𝜃 = 5.19 𝑟𝑎𝑑, 𝑑 = 7.7 𝑚
9. 𝜃 = 2.85 𝑟𝑎𝑑, 𝑟 = 11 𝑚
Given the arc length and central angle 𝜃, find the radius of the circle.
10. 𝜃 = 298°, 𝑠 = 34 𝑚𝑚
11. 𝜃 = 3 𝑟𝑎𝑑, 𝑠 = 4.9 𝑚
12. 𝜃 = 3.1 𝑟𝑎𝑑, 𝑠 = 11.7 𝑐𝑚
Convert each degree measure into radians. Round answers to the 4th decimal place. 13. 14.85°
14. 157.3535°
15. 290.725°
Convert each radian measure into degrees. Round answers to the 4th decimal place.
16. 0.72 radians
17. 2.46 radians
18. 5.11 radians
Find the length of each arc.
19. 𝜃 = 135°, 𝑟 = 12 𝑐𝑚
20. 𝜃 = 4.6 𝑟𝑎𝑑, 𝑑 = 3.7 𝑚
21. 𝜃 = 2.9 𝑟𝑎𝑑, 𝑟 = 39 𝑚𝑚
Given the arc length and central angle 𝜃, find the radius of the circle.
22. 𝜃 = 198°, 𝑠 = 39 𝑐𝑚
23. 𝜃 = 2 𝑟𝑎𝑑, 𝑠 = 9 𝑚
24. 𝜃 = 5.1 𝑟𝑎𝑑, 𝑠 = 19.8 𝑐𝑚
Right Triangle Trigonometry & the Unit Circle Find the exact value of the given expression.
25. cos4𝜋
3
26. sin7𝜋
4
27. sec2𝜋
3
28. tan−5𝜋
6
29. cot15𝜋
4
30. csc−9𝜋
2
Pre-Calculus – Trigonometric Functions ~2~ NJCTL.org
31. Given the terminal point (3
7,
−2√10
7),
find tanθ.
32. Given the terminal point (−5
13,
−12
13),
find cotθ.
33. Knowing cos 𝑥 =2
3 and the terminal point is in the fourth quadrant find sin 𝑥.
34. Knowing cot 𝑥 =4
5 and the terminal point is in the third quadrant find sec 𝑥.
Find the exact value of the given expression.
35. cos5𝜋
3
36. sin3𝜋
4
37. sec4𝜋
3
38. tan−7𝜋
6
39. cot13𝜋
4
40. csc−11𝜋
2
41. Given the terminal point (7
25,
−24
25),
find cotθ
42. Given the terminal point (−4√2
9,
7
9),
find tanθ
43. Knowing sin 𝑥 =7
8 and the terminal point is in the second quadrant find sec 𝑥.
44. Knowing csc 𝑥 = −5
4 and the terminal point is in the third quadrant find cot 𝑥.
Pre-Calculus – Trigonometric Functions ~3~ NJCTL.org
Graphs of Sine and Cosine The sine (red; to the left) and cosine (blue; to the right) waves are shown in the graphs below.
Determine if each statement provided is True or False.
45. sin𝜋
6= sin
5𝜋
6
True
False
46. cos𝜋
3= cos
2𝜋
3
True
False
47. cos2 7𝜋
4=
1
2
True
False
48. sin𝜋
2> sin
3𝜋
2
True
False
49. If sin 𝑘 = −0.75 and cos 𝑘 > 0, what is the exact value of sin(𝑘 − 𝜋)?
a. 0.25
b. –0.25
c. 0.75
d. –0.75
50. If cos 𝑘 = 0.5 and sin 𝑘 < 0, what is the exact value of cos(𝑘 + 5𝜋)?
a. 0.5
b. –0.5
c. 0.866
d. –0.866
State the amplitude, period, and transformations that occur when comparing it to the graph of its
parent trigonometric function (e.g. compare 𝑦 = 2 sin 𝑥 to 𝑦 = sin 𝑥). Draw the graph by hand
and then check it with a graphing calculator.
51. 𝑦 = cos(𝑥 − 𝜋) − 3 52. 𝑦 = 2 sin(𝑥) + 2
Pre-Calculus – Trigonometric Functions ~4~ NJCTL.org
53. 𝑦 = −3
2sin (
𝑥
4) 54. 𝑦 = −1 cos(3𝑥 − 2𝜋) − 1
The sine (red; to the left) and cosine (blue; to the right) waves are shown in the graphs below.
Determine if each statement provided is True or False.
55. sin𝜋
4= sin
7𝜋
4
True
False
56. cos𝜋
3= cos
5𝜋
3
True
False
57. sin2 3𝜋
2= 0
True
False
58. cos𝜋
3< cos
7𝜋
4
True
False
59. If sin 𝑘 = 0.866 and cos 𝑘 < 0, what is the exact value of sin(𝑘 + 4𝜋)?
a. 0.866
b. –0.866
c. 0.5
d. –0.5
60. If cos 𝑘 = −0.707 and sin 𝑘 < 0, what is the exact value of cos(𝑘 − 6𝜋)?
a. 0.707
b. –0.707
c. 0.5
d. –0.5
State the amplitude, period, and transformations that occur when comparing it to the graph of its parent trigonometric function (e.g. compare 𝑦 = 2 sin 𝑥 to 𝑦 = sin 𝑥). Draw the graph by hand and then check it with a graphing calculator.
61. 𝑦 = −3 sin(𝑥) 62. 𝑦 = 2 cos (𝑥
2)
Pre-Calculus – Trigonometric Functions ~5~ NJCTL.org
63. 𝑦 = − sin (𝑥 +𝜋
4) + 4 64. 𝑦 = − cos(6𝑥 − 2𝜋) − 1
The Tangent Function
The tangent function is shown in the graph below on the interval (−𝜋
2,
3𝜋
2). Determine if each
statement is True or False.
65. tan (𝜋
6) = tan (
5𝜋
6)
True False
66. tan (−𝜋
4) + tan (
5𝜋
4) = 0
True False
Pre-Calculus – Trigonometric Functions ~6~ NJCTL.org
67. tan (5𝜋
12) − tan (
𝜋
12) = 2√3
True False
68. tan (𝜋
6) > tan (
2𝜋
3)
True False
69. The segment joining B and A
(7
25, −
24
25) is tangent to the unit circle
at A. Identify the false statement.
a. sin 𝑡 = −24
25
b. tan 𝑡 = −24
7
c. tan(𝑡 − 2𝜋) = −24
7
d. cos(𝑡 + 𝜋) =7
25
70. The segment joining C and D(−0.6, −0.8) is tangent to the unit circle at
D. Identify the false statement.
a. C (−5
3, 0)
b. tan 𝑡 =5
3
c. sin 𝑡 = −0.8 d. CD = tan 𝑡
The tangent function is shown in the graph below on the interval (−𝜋
2,
3𝜋
2). Determine if each
statement is True or False.
71. tan (−𝜋
3) = tan (
4𝜋
3)
True False
72. tan (5𝜋
12) > tan (
4𝜋
3)
True False
73. tan (7𝜋
6) + tan (−
𝜋
6) = 0
True
False
Pre-Calculus – Trigonometric Functions ~7~ NJCTL.org
74. tan (13𝜋
12) + tan (
𝜋
12) = 4
True
False
75. The segment joining G and F (7
25,
24
25)
is tangent to the unit circle at F. Identify the false statement.
a. cos 𝑡 =7
25
b. tan 𝑡 =24
7
c. tan(𝑡 − 3𝜋) = −24
7
d. sin(𝑡 + 𝜋) = −24
25
76. The segment joining J and K
(−15
17,
8
17) is tangent to the unit circle
at J. Identify the false statement.
a. J (17
15, 0)
b. tan 𝑡 = −8
15
c. sin 𝑡 =8
17
d. JK = |tan 𝑡|
The Reciprocal Functions and their Graphs Use the diagram below to determine if each statement is True or False.
77. AC = |sec (4𝜋
3)| = 2
True False
78. AB = |tan (4𝜋
3)| < 1
True False
79. sec (4𝜋
3) cos (
4𝜋
3) = 1
True False
Use the diagram below to determine if each statement is True or False.
80. sin 𝑡 =8
17
True False
81. cos 𝑡 =15
17
True
False
82. sec 𝑡 =17
15
True False
83. tan 𝑡 = −8
15
True False
Pre-Calculus – Trigonometric Functions ~8~ NJCTL.org
84. 𝐴𝑇 = |cot 𝑡| =15
8
True False
85. csc 𝑡 = 2.125 True False
86. The graphs of cosine and secant are shown. Identify the false statement.
a. sec2 (5𝜋
3) = 4
b. sec 2𝜋 = cos 2𝜋
c. The period of sec 𝑥 is 𝜋.
d. If sec 𝑥 = 3, then cos 𝑥 =1
3
87. The graphs of sine and cosecant are shown. Identify the false statement.
a. If csc 𝑥 = 4, then sin 𝑥 =1
4.
b. sin (5𝜋
4) csc (
5𝜋
4) = −1
c. csc 𝑥 = csc(𝑥 − 2𝜋)
d. sin (3𝜋
2) = csc (
3𝜋
2)
88. The graph of cotangent is shown. Identify the false statement.
a. cot2 (𝜋
6) = 3
b. tan (𝜋
3) cot (
2𝜋
3) = −1
c. tan (7𝜋
4) cot (
7𝜋
4) = 1
d. cot (4𝜋
3) cot (
5𝜋
3) = −1
Pre-Calculus – Trigonometric Functions ~9~ NJCTL.org
Use the diagram below to determine if each statement is True or False.
89. AC = csc (𝜋
6) = 2
True False
90. sec2 (𝜋
6) =
3
4
True False
91. sin (𝜋
6) sec (
𝜋
6) = cot (
𝜋
6)
True False
Use the diagram below to determine if each statement is True or False.
92. sin 𝑡 =12
13
True False
93. cos 𝑡 =5
13
True False
94. sec 𝑡 = −2.6 True False
95. 𝐴𝐵 = |tan 𝑡| = 2.4 True False
96. csc 𝑡 = −13
12
True False
97. cot 𝑡 =5
12
True False
98. The graphs of cosine and secant are shown. Identify the false statement.
a. sec2 (5𝜋
6) =
4
3
b. sec (𝜋
2) = cos (
𝜋
2)
c. The period of sec 𝑥 is 2𝜋.
d. If sec 𝑥 = 6, then cos 𝑥 =1
6
99. The graphs of sine and cosecant are shown. Identify the false statement.
a. If csc 𝑥 =4
5, then sin 𝑥 =
5
4.
b. sin (7𝜋
6) csc (
7𝜋
6) = 1
c. The range of csc 𝑥 : (−∞, −1] ∪ [1, ∞)
Pre-Calculus – Trigonometric Functions ~10~ NJCTL.org
d. sin(𝜋) = csc(𝜋)
100. The graphs of tangent and cotangent are shown for one cycle, from 0 and 𝜋. Identify
the false statement.
a. cot (2𝜋
3) > tan (
2𝜋
3)
b. cot (𝜋
6) + cot (
5𝜋
6) = 0
c. cot (𝜋
3) > tan (
𝜋
3)
d. If cot 𝑥 = tan 𝑥 = −1, then 𝑥 =3𝜋
4
Graphs of Composite Trigonometric Functions
State the amplitude, period, and transformations that occur when comparing the new function to the graph of its parent trigonometric function (e.g. compare 𝑦 = 2 tan 𝑥 to 𝑦 = tan 𝑥). Draw the graph by hand and then check it with a graphing calculator.
101. 𝑦 =1
4sec 𝑥 102. 𝑦 = csc (𝑥 − 𝜋)
103. 𝑦 = −1
2cot 𝑥 104. 𝑦 = 1 + sec (𝑥 + 𝜋)
Pre-Calculus – Trigonometric Functions ~11~ NJCTL.org
105. 𝑦 =1
3tan(𝑥) + 1 106. 𝑦 = −3 tan(𝑥) − 1
State the amplitude, period, and transformations that occur when comparing the new function to the graph of its parent trigonometric function (e.g. compare 𝑦 = 2 tan 𝑥 to 𝑦 = tan 𝑥). Draw the graph by hand and then check it with a graphing calculator.
107. 𝑦 = csc (𝑥
3) 108. 𝑦 = tan(𝑥 + 𝜋) − 1
109. 𝑦 =1
2sec(2𝑥) 110. 𝑦 = 2 cot (𝑥 +
𝜋
4)
111. 𝑦 = 2 tan(𝑥) − 1 112. 𝑦 = 2 + 2 sec 4𝑥
Pre-Calculus – Trigonometric Functions ~12~ NJCTL.org
Inverse Trigonometric Functions Determine the exact value of each inverse trig function within the indicated interval.
113. arctan (√3
3), (−
𝜋
2,
𝜋
2)
114. arcsin (−1
2) , [−
𝜋
2,
𝜋
2]
115. cos−1 (−√2
2) , [0, 𝜋]
116. sin−1 (√3
2) , [−
𝜋
2,
𝜋
2]
117. arccos(0) , [0, 𝜋]
118. tan−1(0) , (−
𝜋
2,
𝜋
2)
119. The figure is the unit circle with an arc that measures 𝜋
3, as shown. Which statement is
true?
a. sin−1 (𝜋
3) =
√3
2
b. arcsin (2√3
3) =
𝜋
3
c. sin (√3
2) =
𝜋
3
d. arcsin (√3
2) =
𝜋
3
120. The completed graphs of the restricted sine and inverse sine are shown. Which statement is false?
a. sin−1(−1) = −𝜋
2
b. sin (𝜋
6) > sin−1 (
𝜋
6)
c. sin (−𝜋
4) > sin−1 (−
𝜋
4)
d. sin (𝜋
2) < sin−1(1)
121. The figure is the unit circle with an arc t that terminates at 𝑃 (−8
17,
15
17). Which
statement is false?
a. cos 𝑡 = −8
17
b. cos−1 (−8
17) = 𝑡
c. sin−1 (15
17) = 𝑡
Pre-Calculus – Trigonometric Functions ~13~ NJCTL.org
d. sin 𝑡 =15
17
122. The figure is the unit circle with an arc t that terminates at 𝑄(0.8, −0.6). Which
statement is false? a. 𝑃𝑅 = |tan 𝑡| = 0.75
b. 𝑡 = arctan (−3
4)
c. sin 𝑡 < cos 𝑡 d. sin 𝑡 cos 𝑡 = 0.48
Determine the exact value of each inverse trig function within the indicated interval.
123. arctan(√3), (−𝜋
2,
𝜋
2)
124. arcsin (1
2) , [−
𝜋
2,
𝜋
2]
125. cos−1 (−√3
2) , [0, 𝜋]
126. sin−1 (√2
2) , [−
𝜋
2,
𝜋
2]
127. arccos(−1) , [0, 𝜋]
128. tan−1(1) , (−
𝜋
2,
𝜋
2)
129. The completed graphs of the restricted cosine and inverse cosine are shown. Which statement is false?
a. cos−1(−1) = 𝜋
b. cos (𝜋
6) < cos−1 (
𝜋
6)
c. cos (𝜋
12) < cos−1 (
𝜋
12)
d. cos (𝜋
3) > cos−1 (
𝜋
3)
130. The figure is the unit circle with an arc that measures 5𝜋
6, as shown. Which statement
is true?
a. cos−1 (5𝜋
6) = −
√3
2
b. sin (5𝜋
6) =
1
2
c. arcsin (1
2) =
5𝜋
6
d. cos (−√3
2) =
5𝜋
6
131. The figure is the unit circle with an arc t that terminates at 𝑃 (7
25,
24
25). Which statement
is false?
Pre-Calculus – Trigonometric Functions ~14~ NJCTL.org
a. cos 𝑡 =7
25
b. cos−1 (7
25) = 𝑡
c. sin−1 (24
25) = 𝑡
d. tan−1 (7
24) = 𝑡
132. The figure is the unit circle with an arc 𝑡 = −𝜋
4 that terminates at 𝑄. Which statement
is false?
a. sin (−𝜋
4) = sin (
𝜋
4)
b. arctan(−1) = −𝜋
4
c. 𝑃𝑅 = |tan (−𝜋
4)| = 1
d. cos (−𝜋
4) = cos (
𝜋
4)
Pre-Calculus – Trigonometric Functions ~15~ NJCTL.org
Modeling using Trigonometry
Solve each word problem.
133. The table gives the normal daily temperatures in Philadelphia P in degrees Fahrenheit
for month t with t = 1 corresponding to January
The model for these temperatures is given by
𝑃(𝑡) = 63.17 + 22.75 sin (𝜋𝑡
6+ 4.21).
a. Use a graphing utility to graph the data points & the model for the temperatures in Philadelphia. How well does the model fit?
b. Find the average annual temperature in Philadelphia. Which term of the equation is closely related to your average?
c. What is the period in this model? What does it stand for?
134. A mass suspended from a spring is compressed a distance of 4 cm above its rest
position, as shown in the figure. The mass is released after time t = 0 and allowed to
oscillate. It is observed that the mass reaches its lowest point 1
2 second after it is
released. Find an equation that describes the motion of the mass.
135. A Ferris wheel has a radius of 9 m, and the bottom of the wheel passes 1 m above the
ground. The Ferris wheel makes one complete revolution every 20 s, and a person
riding the Ferris wheel is at a minimum value when 𝑡 = 0. Find an equation that gives the height above the ground of a person on the Ferris wheel as a function of time.
Pre-Calculus – Trigonometric Functions ~16~ NJCTL.org
Solve each word problem.
136. The table gives the normal daily temperatures in Honolulu H in degrees Fahrenheit for
month t with t = 1 corresponding to January
The model for these temperatures is given by
𝐻(𝑡) = 84.40 + 4.28 sin (𝜋𝑡
6+ 3.86).
a. Use a graphing utility to graph the data points & the model for the temperatures in Honolulu. How well does the model fit?
b. Find the average annual temperature in Honolulu. Which term of the equation is closely related to your average?
c. What is the period in this model? What does it stand for?
137. A mass suspended from a spring is pulled down a distance of 0.6 m from it’s rest
position, as shown in the figure. The mass is released after time t = 0 and allowed to
oscillate. If the mass returns to its rest position after 1 second, find an equation that
describes its motion.
138. A Ferris wheel has a radius of 11 m, and the bottom of the wheel passes 1 m above
the ground. The Ferris wheel makes one complete revolution every 30 s, and a person
riding the Ferris wheel is at a minimum value when 𝑡 = 0. Find an equation that gives the height above the ground of a person on the Ferris wheel as a function of time.
Pre-Calculus – Trigonometric Functions ~17~ NJCTL.org
Unit Review Multiple Choice: 1. How many radians is 195°?
a. 8𝜋
9 radians
b. 8𝜋
3 radians
c. 7𝜋
6 radians
d. 13𝜋
12 radians
2. Given that the terminal side of an angle in standard position goes through the point
𝑃(−20, −21), which of the following statements is false?
a. tan 𝜃 = −21
20
b. csc 𝜃 = −29
21
c. sec 𝜃 = −20
29
d. cot 𝜃 =20
21
3. Consider the terminal point and 𝑡 in the diagram. Four of the statements are true; one is
false. Which statement is false?
a. tan(𝑡) =5
12
b. sin(−𝑡) = −5
13
c. cos(𝑡 + 𝜋) =12
13
d. cos(𝑡 + 𝜋) = −5
13
4. One cycle of the sine wave is shown. Three of the statements are true, and one is false.
Which statement is false?
a. sin5𝜋
12= sin
7𝜋
12
b. sin𝜋
4= sin
3𝜋
4
c. sin𝜋
3= sin
5𝜋
3
d. sin7𝜋
6= sin
11𝜋
6
5. If cos 𝜃 = −0.96, and sin 𝜃 < 0, what is the exact value of sin (𝜃 + 𝜋)?
a. –0.28
b. 0.28
c. 0.96
d. –0.96
Pre-Calculus – Trigonometric Functions ~18~ NJCTL.org
6. Rank in order from smallest to largest.
I. sin13𝜋
12 II. sin
11𝜋
6 III. sin
2𝜋
3
a. I < II < III
b. I < III < II
c. II < III < I
d. II < I < III
7. More than one cycle of the cosine wave is shown. Three of the statements are true, and
one is false. Which statement is false?
a. cos𝜋
6= cos
7𝜋
6
b. cos2𝜋
3< cos
5𝜋
3
c. cos𝜋
3+ cos
4𝜋
3= 0
d. cos7𝜋
12> cos
7𝜋
6
8. More than one cycle of the sine and cosine waves are shown. Three of the statements are
true, and one is false. Which statement is false?
a. sin (𝜋
6) + cos (
𝜋
3) = 1
b. cos3𝜋
4= sin
5𝜋
4
c. sin17𝜋
12= cos
13𝜋
12
d. sin𝜋
3> cos
11𝜋
12
Pre-Calculus – Trigonometric Functions ~19~ NJCTL.org
9. Which graph represents 𝑦 =1
2cos(𝑥 − 𝜋) − 1?
a.
b.
c.
d.
10. Which graph represents 𝑦 = cos(2𝑥) 5?
a.
b.
Pre-Calculus – Trigonometric Functions ~20~ NJCTL.org
c.
d.
11. A graph of the tangent function is shown below in the interval [−𝜋
2,
5𝜋
2] with the given points.
Which statement is false?
a. tan3𝜋
4= tan
7𝜋
4
b. tan𝜋
3= tan
4𝜋
3
c. tan5𝜋
6+ tan
7𝜋
6= 0
d. tan13𝜋
12> tan
11𝜋
12
12. The segment joining R and C (–8/17, –15/17) is tangent to the unit circle at point C. Identify
the false statement. a. 𝐶𝑅 = |tan 𝑡|
b. sin 𝑡 = −15
17
c. tan 𝑡 =8
15
d. cos 𝑡 = −8
17
Pre-Calculus – Trigonometric Functions ~21~ NJCTL.org
13. There is a first quadrant terminal point on the unit circle where sin 𝑡 = 3 cos 𝑡. Identify the
false statement.
a. cot 𝑡 =1
3
b. sec 𝑡 = √10
c. sin 𝑡 cos 𝑡 =3
10
d. sin 𝑡 =√10
10
14. The graphs of cosine and secant are shown below. Identify the false statement.
a. sec(𝜋) = cos(𝜋)
b. cos4𝜋
3= −
1
2
c. sec2 (𝜋
6) = 4
d. cos 2π − sec 𝜋 = 2
15. The graphs of tangent and cotangent are shown below. Identify the false statement.
a. If tan 𝑥 = cot 𝑥 = 1, then 𝑥 =𝜋
4.
b. tan𝜋
6> cot
𝜋
6
c. tan𝜋
3= cot
𝜋
6
d. cot𝜋
12+ cot
11𝜋
12= 0
16. Which graph represents 𝑓(𝑥) = 2 sec 𝑥 + 1 on the interval [0, 2𝜋]?
a.
b.
Pre-Calculus – Trigonometric Functions ~22~ NJCTL.org
c.
d.
17. The complete graphs of the restricted cosine and inverse cosine are shown below.
Which statement is false?
a. cos−1 1 = cos 0
b. cos𝜋
2= cos−1 1
c. 𝑐𝑜𝑠−10.5 > cos𝜋
6
d. cos𝜋
6> cos−1 (−
√3
2)
18. The figure below is the unit circle with an arc t that
terminates at 𝑃 (−8
17,
15
17). Which statement is false?
a. cos(𝑡 + 𝜋) =8
17
b. sin(𝑡 + 𝜋) = −15
17
c. cos−1 (8
17) = 𝑡
d. sin−1 (−15
17) = 𝑡
19. A piston moves up and down in a cylinder. In this simulation, it takes 3 seconds for the
piston to complete one cycle. If the low position of the piston is 𝑦 = 2 𝑐𝑚 and the high
position of the piston is 𝑦 = 22 𝑐𝑚. Which function would be used to model this situation?
a. 𝑦 = 10 sin2𝜋
3𝑡 + 12
b. 𝑦 = 10 sin3𝜋
2𝑡 + 12
c. 𝑦 = 12 sin2𝜋
3𝑡 + 10
d. 𝑦 = 12 sin3𝜋
2𝑡 + 10
Pre-Calculus – Trigonometric Functions ~23~ NJCTL.org
Extended Response:
20. A mass suspended from a spring is compressed a distance of 6 cm above its rest
position, as shown in the figure. The mass is released after time t = 0 and allowed to
oscillate. It is observed that the mass reaches its lowest point 1
4 second after it is
released.
a. What is the amplitude of the motion of the mass?
b. What is the period of the motion of the mass?
c. Write a function that describes the motion of the mass. 𝑦 = 4 cos(2𝜋𝑡)
21. A Ferris wheel has a diameter of 36 m, and the bottom of the wheel passes 2 m above
the ground. The Ferris wheel makes one complete revolution every 70 s, and a person
riding the Ferris wheel is at a minimum value when 𝑡 = 0.
a. If a person riding the Ferris wheel is at a minimum value when 𝑡 = 0, find an
equation that gives the person’s height above the ground as a function of time.
b. After how many seconds will a person reach a height of 30m above the ground for the first time?
Pre-Calculus – Trigonometric Functions ~24~ NJCTL.org
Answer Key
1. 0.6000 rad
2. 3.0802 rad
3. 3.9419 rad
4. 14.3239°
5. 76.7763°
6. 245.2259°
7. 𝑠 =47𝜋
4𝑐𝑚 ≈ 36.9137 𝑐𝑚
8. s = 19.9815 m
9. s = 31.35 m
10. 𝑟 =3060
149𝜋𝑚𝑚 ≈ 6.5371 𝑚𝑚
11. r = 1.6333 m
12. r = 3.7742 cm
13. 0.2592 rad
14. 2.7463 rad
15. 5.0471 rad
16. 41.2530°
17. 140.9476°
18. 292.7814°
19. 𝑠 = 9𝜋 𝑐𝑚 ≈ 28.2743 𝑐𝑚
20. s = 8.51 m
21. s = 113.1 mm
22. 𝑟 =390
11𝜋𝑐𝑚 ≈ 11.2855 𝑐𝑚
23. r = 4.5 m
24. r = 3.8824 cm
25. −1
2
26. −√2
2
27. −2
28. √3
3
29. −1
30. −1
31. tan 𝜃 = −2√10
3
32. cot 𝜃 =5
12
33. sin 𝑥 = −√5
3
34. cot 𝑥 = −√41
4
35. 1
2
36. √2
2
37. −2
38. −√3
3
39. 1
40. 1
41. cot 𝜃 = −7
24
42. tan 𝜃 = −7√2
8
43. sec 𝑥 = −8√15
15
44. cot 𝑥 =3
4
45. True
46. False
47. True
48. False
49. C
50. B
51. Amplitude: 1
Period: 2𝜋
Transformations:
horizontal shift left 𝜋 units
vertical shift down 3 units
52. Amplitude: 2
Period: 2𝜋
Transformations:
vertical stretch with a factor of 2
vertical shift up 2 units
Pre-Calculus – Trigonometric Functions ~25~ NJCTL.org
53. Amplitude: 3
2
Period: 8𝜋
Transformations:
horizontal stretch with a factor of 1
4
vertical stretch with a factor of 3
2
reflection across the x-axis
54. Amplitude: 1
Period: 2𝜋
3
Transformations:
horizontal shift right 2𝜋 units
horizontal shrink with a factor of 3
reflection about the x-axis
vertical shift down 1 unit
55. False
56. True
57. False
58. True
59. A
60. B
61. Amplitude: 3
Period: 2𝜋
Transformations:
vertical stretch with a factor of 3
reflection about the x-axis
62. Amplitude: 2
Period: 4𝜋
Transformations:
horizontal stretch with a factor of ½
vertical stretch with a factor of 2
63. Amplitude: 1
Period: 2𝜋
Transformations:
horizontal shift left 𝜋
4 units
reflection about the x-axis
vertical shift up 4 units
64. Amplitude: 1
Period: 𝜋
3
Transformations:
horizontal shift right 2𝜋 units
horizontal shrink with a factor of 6
reflection about the x-axis
Pre-Calculus – Trigonometric Functions ~26~ NJCTL.org
vertical shift down 1 unit
65. False
66. True
67. True
68. True
69. D
70. B
71. False
72. True
73. True
74. False
75. C
76. A
77. True
78. False
79. True
80. False
81. True
82. True
83. True
84. True
85. False
86. C
87. B
88. D
89. True
90. False
91. False
92. True
93. False
94. True
95. True
96. False
97. False
98. B
99. D
100. C
101. Amplitude: 1
4
Period: 2𝜋
Transformations:
vertical shrink with a factor of 1
4
102. Amplitude: 1
Period: 2𝜋
Transformations:
vertical shrink with a factor of 1
3
vertical shift up 1 unit
103. Amplitude: 1
2
Period: 𝜋
Transformations:
vertical shrink with a factor of 1
2
reflection about the x-axis
104. Amplitude: 1
Period: 2𝜋
Transformations:
horizontal shift left 𝜋 units
Pre-Calculus – Trigonometric Functions ~27~ NJCTL.org
vertical shift up 1 unit
105. Amplitude: 1
3
Period: 𝜋
Transformations:
vertical shrink with a factor of 1
3
vertical shift up 1 unit
106. Amplitude: 3
Period: 𝜋
Transformations:
vertical stretch with a factor of 3
reflection about the x-axis
vertical shift down 1 unit
107. Amplitude: 1
Period: 6𝜋
Transformations:
horizontal stretch with a factor of 1
3
108. Amplitude: 1
Period: 𝜋
Transformations:
horizontal shift left 𝜋 units
vertical shift down 1 unit
109. Amplitude: 1
2
Period: 𝜋
2
Transformations:
horizontal shrink with a factor of 2
vertical shrink with a factor of 1
2
110. Amplitude: 2
Period: 𝜋
Transformations:
horizontal shift left 𝜋
4 units
Pre-Calculus – Trigonometric Functions ~28~ NJCTL.org
vertical stretch with a factor of 2
111. Amplitude: 2
Period: 𝜋
Transformations:
vertical stretch with a factor of 2
vertical shift down 1 units
112. Amplitude: 2
Period: 𝜋
2
Transformations:
horizontal shrink with a factor of 4
vertical stretch with a factor of 2
vertical shift up 2 units
113. 𝜋
6
114. −𝜋
6
115. 𝜋
4
116. 𝜋
3
117. 𝜋
2
118. 0
119. D
120. B
121. C
122. D
123. 𝜋
3
124. 𝜋
6
125. 5𝜋
6
126. 𝜋
4
127. 𝜋
128. 𝜋
4
129. D
130. B
131. D
132. A
133.
a. The model function fits the data
pretty well.
b. Avg annual temp. = 63.17℉
This is the midline of the
trigonometric function.
c. The period ranges from 1 to 12.
These numbers represent the
months in a year.
134. 𝑦 = 4 cos(2𝜋𝑡)
135. 𝑦 = 9 sin (𝜋
10(𝑡 − 5)) + 10 or
𝑦 = −9 cos (𝜋
10𝑡) + 10
136.
a. The model function fits the data
pretty well.
b. Avg annual temp. = 84.4℉
This is the midline of the
trigonometric function.
c. The period ranges from 1 to 12.
These numbers represent the
months in a year.
137. 𝑦 = −0.6 cos (𝜋
2𝑡)
138. 𝑦 = 11 sin (𝜋
15(𝑡 − 7.5)) + 12 or
𝑦 = −11 cos (𝜋
15𝑡) + 12
Pre-Calculus – Trigonometric Functions ~29~ NJCTL.org
Unit Review
1. D 2. C 3. A 4. C 5. B 6. D 7. A 8. B 9. A 10. D
11. B 12. C 13. D 14. C 15. B 16. B 17. D 18. C 19. A
20.
a. 6 𝑐𝑚 b. 1 𝑠𝑒𝑐𝑜𝑛𝑑
c. 𝑦 = 6cos (𝜋𝑡) 21.
a. 𝑦 = −18 cos (𝜋
35𝑡) + 20
or 𝑦 = −18 sin (𝜋
35(𝑡 − 17.5)) + 20
b. 24.062 𝑠𝑒𝑐𝑜𝑛𝑑𝑠