PRECALCULUS UNIT 4: TRIGONOMETRY PRACTICE PROBLEMS · 2019-11-14 · Pre-Calculus –...

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


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.


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


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)


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


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


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


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



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


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, ∞)


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)


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


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


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?


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)


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.


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


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.


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


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


9. Which graph represents 𝑦 =1

2cos(𝑥 − 𝜋) − 1?

a.

b.

c.

d.

10. Which graph represents 𝑦 = cos(2𝑥) 5?

a.

b.


c.

d.

11. A graph of the tangent function is shown below in the interval [−𝜋

2,

5𝜋

2] with the given points.


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


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.


c.

d.

17. The complete graphs of the restricted cosine and inverse cosine are shown below.


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


Extended Response:

20. A mass suspended from a spring is compressed a distance of 6 cm above its rest


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?


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


53. Amplitude: 3

2

Period: 8𝜋

Transformations:

horizontal stretch with a factor of 1

4


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:



62. Amplitude: 2

Period: 4𝜋

Transformations:

horizontal stretch with a factor of ½


63. Amplitude: 1

Period: 2𝜋

Transformations:

horizontal shift left 𝜋

4 units



64. Amplitude: 1

Period: 𝜋

3

Transformations:

horizontal shift right 2𝜋 units





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:


3

vertical shift up 1 unit

103. Amplitude: 1

2

Period: 𝜋

Transformations:


2


104. Amplitude: 1

Period: 2𝜋

Transformations:




105. Amplitude: 1

3

Period: 𝜋

Transformations:


3


106. Amplitude: 3

Period: 𝜋

Transformations:




107. Amplitude: 1

Period: 6𝜋

Transformations:

horizontal stretch with a factor of 1

3

108. Amplitude: 1

Period: 𝜋

Transformations:



109. Amplitude: 1

2

Period: 𝜋

2

Transformations:



2

110. Amplitude: 2

Period: 𝜋

Transformations:

horizontal shift left 𝜋

4 units



111. Amplitude: 2

Period: 𝜋

Transformations:


vertical shift down 1 units

112. Amplitude: 2

Period: 𝜋

2

Transformations:




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


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

PRECALCULUS UNIT 4: TRIGONOMETRY PRACTICE PROBLEMS · 2019-11-14 · Pre-Calculus –...

Documents

Transcript of PRECALCULUS UNIT 4: TRIGONOMETRY PRACTICE PROBLEMS · 2019-11-14 · Pre-Calculus –...