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Alg 2-Trig Functions ~1~ NJCTL.org

Converting Degrees and Radians – Class Work

Convert the following degree measures to radians and radian measures to degrees. Sketch

each angle.

1. 2𝜋

3

2. 35°

3. 225°

4. π

5

5. 150°

6. 14𝜋

9

7. 310°

8. 10π

7

Converting Degrees and Radians – Homework

Convert the following degree measures to radians and radian measures to degrees. Sketch

each angle.

9. 5𝜋

3

10. 75°

11. 200°

12. π

6

13. 175°

14. 17𝜋

9

15. 350°

16. 9π

7

Co-terminal Angles – Classwork

Name one positive angle and one negative angle that is co-terminal with the given angle.

17. 2𝜋

3

18. 35°

19. 225°

20. π

5

21. 150°

22. 14𝜋

9

23. 310°

24. 10π

7


Co-terminal Angles – Homework

Name one positive angle and one negative angle that is co-terminal with the given angle.

25. 5𝜋

3

26. 75°

27. 200°

28. π

6

29. 175°

30. 17𝜋

9

31. 350°

32. 9π

7

Arc Length and Sector Area - Classwork

Round all lengths to the nearest tenth.

For problems 33 - 36 below, 𝜃 is the radian measure of a central angle that intercepts an arc of

length 𝑠 in a circle with a radius 𝑟.

33. If 𝑠 = 10 and 𝑟 = 5, find 𝜃.

34. If 𝜃 =𝜋

3 and 𝑟 = 6, find 𝑠.

35. If 𝑠 = 5.4 𝑎𝑛𝑑 𝑟 = 1.8, 𝑓𝑖𝑛𝑑 𝜃.

36. If 𝑠 = 15 𝑎𝑛𝑑 𝜃 =3𝜋

4, 𝑓𝑖𝑛𝑑 𝑟.

37. If 𝑟 = 9 𝑎𝑛𝑑 𝜃 = 3𝜋, 𝑓𝑖𝑛𝑑 𝑠.

38. If 𝜃 = 6𝜋 𝑎𝑛𝑑 𝑠 = 9, 𝑓𝑖𝑛𝑑 𝑟.

39. Find the area of a sector with radius 5 inches and central angle 𝜃 =𝜋

12.

40. Find the area of a sector with radius 6 cm and central angle 𝜃 = 150°.

41. Find the radius of a sector with area 45 sq in and central angle 𝜃 =5𝜋

12.

42. The central angle of a circle has a measure of 7 radians and it intercepts an arc whose length is 9 meters. What is the length in meters of the radius of the circle?

43. The minute hand of a clock makes what angle as it moves from 6:15 to 6:45? If the length of the intercepted arc is 15 inches, what is the length of the minute hand?


44. The wheels of a car have a diameter of 36 inches. The wheels of a scooter have a diameter of 10 inches. If each wheel makes one complete rotation, do the car and the scooter travel the same distance? If no, which travels farther, and by how much?

45. A wedge of a round cake is cut to be one-sixth of the cake. If the diameter of the cake is 10 inches, what is the length of the intercepted arc of the top of the cake?

46. Billy Bob got 1/3 of a 6-inch pie and Sally Sue got ¼ of an 8-inch pie. Who got more pie and by what percent?

47. Go back to the dartboard problem on slide 30. What is the probability that a dart thrown at random at the board lands in the black space?

Arc Length and Sector Area - Homework

For problems 43 - 46 below, 𝜃 is the radian measure of a central angle that intercepts an arc of length 𝑠 in a circle with a radius 𝑟.

48. If 𝑠 = 8 and 𝑟 = 9, find 𝜃.

49. If 𝜃 =5𝜋

3 and 𝑟 = 6, find 𝑠.

50. If 𝑠 = .001 𝑎𝑛𝑑 𝑟 = .00025, 𝑓𝑖𝑛𝑑 𝜃.

51. If 𝑠 = 20 𝑎𝑛𝑑 𝜃 =9𝜋

4, 𝑓𝑖𝑛𝑑 𝑟.

52. If 𝑟 = 1.5 𝑎𝑛𝑑 𝜃 = 𝜋, 𝑓𝑖𝑛𝑑 𝑠.

53. If 𝜃 = 4𝜋 𝑎𝑛𝑑 𝑠 = 18, 𝑓𝑖𝑛𝑑 𝑟.

54. Find the area of a sector with radius 11 inches and central angle 𝜃 =𝜋

9.

55. Find the area of a sector with radius 9 cm and central angle 𝜃 = −140°.

56. Find the radius of a sector with area 12 sq in and central angle 𝜃 =3𝜋

4.


57. If a circle has a radius of 6 inches and a central angle intercepts an arc of 11 inches,

what is the radian measure of the central angle?

58. The minute hand of a clock makes what angle as it moves from 8:05 to 8:57? If the

length of the intercepted arc is 18 inches, what is the length of the minute hand?

59. The wheel of a unicycle has a radius of 24 inches. The wheels of a tricycle have a radius

of 16 inches. If each wheel makes one complete rotation, do the car and the scooter

travel the same distance? If no, which travels farther, and by how much?

60. A wedge of pie is cut to be one-seventh of the pie. If the length of the intercepted arc of

the top of the pie is 4.3 inches, what is the diameter of the pie?

61. Billy Bob got 3 8⁄ of an 18-inch pizza pie and Sally Sue got 4 9⁄ of a 16-inch pie. Who got

more pizza and by what percent?

Unit Circle – Class Work

62. Given the terminal point (3

7,

−2√10

7) find tanθ and 𝜃.

63. Given the terminal point (−5

13,

−12

13) find cot 𝜃 and 𝜃.

64. Given cos 𝜃 = 2

3 and the terminal point in the fourth quadrant, find sin 𝜃.

65. Given cot 𝜃 = 4

5 and the terminal point in the third quadrant, find sec 𝜃.

For problems 53 - 56, for each given function value, find the values of the other five trig

functions.


66. sin 𝜃 = −1

4 and the terminal point is in the fourth quadrant.

67. tan 𝜃 = −2 and the terminal point is in the second quadrant.

68. csc 𝜃 =8

5 and the terminal point is in the second quadrant.

69. sec 𝜃 = 3 and the terminal point is in the fourth quadrant.

State the quadrant in which 𝜃 lies:

70. sin 𝜃 > 0, cos 𝜃 > 0

71. sin 𝜃 < 0, tan 𝜃 > 0

72. csc 𝜃 < 0, sec 𝜃 > 0

73. sin 𝜃 > 0, cot 𝜃 > 0

Find the exact value of the given expression.

74. cos4π

3

75. sin7π

4

76. sec2π

3

77. tan-5π

6

78. cot15π

4

79. csc-9π

2

Find the exact value of the sine, cosine and tangent of the given angle.

80. 4𝜋

3

81. –𝜋

2

82. 11𝜋

4

83. 210°

84. -315°


Unit Circle – Homework

85. Given the terminal point (7

25,

−24

25) find cotθ and 𝜃.

86. Given the terminal point (−4√2

9,

7

9) find tanθ and 𝜃.

87. Given sin 𝜃= 7

8 and the terminal point in the second quadrant, find sec 𝜃.

88. Given csc 𝜃 = 5

−4 and the terminal point in the third quadrant find cot 𝜃.

For problems 68 - 71, for each given function value, find the values of the other five trig

functions.

89. sin 𝜃 =9

41 and the terminal point is in the second quadrant.

90. cot 𝜃 = −3 and the terminal point is in the second quadrant.

91. cos 𝜃 = −3

5 and the terminal point is in the third quadrant.

92. sin 𝜃 = 0.7 and the terminal point is in the second quadrant.


State the quadrant in which 𝜃 lies:

93. sin 𝜃 > 0, cos 𝜃 < 0

94. sin 𝜃 < 0, tan 𝜃 < 0

95. csc 𝜃 > 0, sec 𝜃 > 0

96. sin 𝜃 < 0, cot 𝜃 < 0

Find the exact value of the given expression.

97. cos5π

3

98. sin3π

4

99. sec4π

3

100. tan−7π

6

101. cot13π

4

102. csc−11𝜋

2

Find the exact value of the sine, cosine and tangent of the given angle.

103. 8𝜋

3

104. 5𝜋

4

105. −7𝜋

6

106. 690°

107. -240°

Graphing Classwork

Use the functions below to answer questions 108 – 111.

a. 𝑦 = 2 cos 𝑥 b. 𝑦 = −2 sin 2𝑥

c. 𝑦 = −3 sin𝑥

2+ 1 d. 𝑦 = cos (𝑥 −

𝜋

3)

e. 𝑦 = sin (𝑥 +𝜋

4) f. 𝑦 = 2 cos (2𝑥 −

𝜋

3)

g. 𝑦 = −4 sin(0.5𝑥 + 𝜋) + 1

108. Find the amplitude of each function.


109. Find the period of each function.

110. Find the phase shift of each function.

111. Find the vertical shift of each function.

112. Sketch one cycle of each function on graph paper.

113. Is the graph of 𝑦 = cos 𝑥 is the same as the graph of 𝑦 = sin (𝑥 −𝜋

2)? Justify your

answer.

For each graph below, name the amplitude, period and vertical shift. Write an equation to

represent each graph.

114. 115.

116. 117.

State the amplitude, period, phase shift, and vertical shift for each function. Draw the graph by

hand and then check it with a graphing calculator.

118. 𝑦 = 2 cos (𝑥 +𝜋

3) + 1


119. 𝑦 = −3 cos(4𝑥 − 𝜋) − 2

120. 𝑦 = sin (2

3(𝑥 +

𝜋

6)) + 3

121. 𝑦 = −1 cos(3𝑥 − 2𝜋) − 1

122. 𝑦 =2

3cos(4𝑥 − 2𝜋) + 2

123. The musical note A above middle C on a piano makes a sound that can be modeled by

the sine wave 𝑦 = sin(880𝜋𝑥), where x represents time in seconds, and y represents the

sound pressure. What is the period of this function?

124. A row boat in the ocean oscillates up and down with the waves. The boat moves a total

of 10 feet from its low point to its high point and then returns to its low point every 11

seconds. Write an equation to represent the boat’s position y at time t, if the boat is at its low

point at t = 0.

Graphing – Homework

Use the functions below to answer questions 124 – 127.

a. 𝑦 = −3 cos 𝑥 b. 𝑦 = −2 sin 2𝑥

c. 𝑦 = − sin𝑥

6 d. 𝑦 = cos (𝑥 +

2𝜋

3)

e. 𝑦 = −2 sin (𝑥 +𝜋

4) f. 𝑦 = 4 cos (𝑥 −

𝜋

3) − 2

g. 𝑦 = −2 sin(𝑥 + 3𝜋) + 5

125. Find the amplitude of each function.

126. Find the period of each function.

127. Find the phase shift of each function.


128. Find the vertical shift of each function.

129. Sketch one cycle of each function on graph paper.

State the amplitude, period, phase shift, and vertical shift for each function. Draw the graph by

hand and then check it with a graphing calculator.

130. 𝑦 = −4 cos (1

2(𝑥 −

𝜋

3)) + 2

131. 𝑦 = −2 cos(4𝑥 − 3𝜋) − 3

132. 𝑦 = 2 sin (1

4(𝑥 +

𝜋

2)) + 1

133. 𝑦 = −1 cos(6𝑥 − 2𝜋) − 1

134. 𝑦 =3

2cos(4𝑥 − 3𝜋) − 2

135. The musical notes C# (C sharp) and E can be modeled by the sine waves 𝑦 =

sin(1100𝜋𝑥), and 𝑦 = sin(1320𝜋𝑥) respectively , where x represents time in seconds, and

y represents the sound pressure. What are the periods of these functions?

136. A swimmer on a raft in the ocean oscillates up and down with the waves. The raft moves

a total of 7 feet from its low point to its high point and then returns to its low point every 8

seconds. Write an equation to represent the raft’s position y at time t, if the raft is at its

low point at t = 0.


Trigonometric Identities – Class Work

Simplify the expression

137. csc 𝑥 tan 𝑥 138. cot 𝑥 sec 𝑥 sin 𝑥 139. sin x (csc x − sin x)

140. (1 + cot2x)(1 − cos2x) 141. 1 − tan2x ÷ sec2 𝑥

142. (sin x − cos x)2 143. cot2x

1−sin2x

144. cos 𝑥

sec 𝑥+tan 𝑥 145. sin 𝑥 tan 𝑥 + cos 𝑥

Verify the Identity

146. (1 − sin 𝑥)(1 + sin 𝑥) = cos2 x 147. tan 𝑥 cot 𝑥

sec 𝑥= cos 𝑥

148. (1 − cos2x)(1 + tan2x) = tan2x 149. 1

sec x+tan x+

1

sec x−tan x= 2 sec x


Trigonometric Identities – Homework

Simplify the expression

150. (tan x + cot x )2 151. 1−sin x

cos x+

cos x

1−sin x 152.

cos x−cos y

sin x+sin y+

sin x−sin y

cos x+cos y

153. 1

sin 𝑥−

1

csc 𝑥 154.

1+sec2x

1+tan2x

155. sin2x

tan2x+

cos2x

cot2x 156.

𝑡𝑎𝑛2𝑥

1+𝑡𝑎𝑛2𝑥

157. cos x

sec x+

sin x

csc x 158.

1+sec2x

1+tan2x+

cos2x

cot2x

Verify the Identity

159. 𝑐𝑜𝑠2𝑥 − 𝑠𝑖𝑛2𝑥 = 1 − 2𝑠𝑖𝑛2𝑥 160. tan 𝑥 cos 𝑥 csc 𝑥 = 1

161. 1+cot x

csc x= sin x + cos x 162.

cos x csc x

cot x= 1


Unit Review

Multiple Choice

1. How many degrees is 4π

9?

a. 160°

b. 110°

c. 80°

d. 62°

2. Which angle is 11𝜋

3?

a. c.

b. d.

3. Which of the following angles is/are co-terminal with 170° (choose all correct answers)?

a. 340°

b. 190°

c. -190°

d. 530°

4. Which is larger and by how much: an angle of 258°, or an angle of 10𝜋

7 radians?

a. 258° by 6

7°

b. 258° by 6

7 radian

c. 10𝜋

7 radians by

1

7°

d. 10𝜋

7 radians by

6

7°

5. The central angle of a circle has a measure of 5𝜋

4 radians and it intercepts an arc whose

length is 5 meters. What is the approximate length in meters of the radius of the circle?

a. 19.6 m

b. 2.0 m

c. 1.3 m

d. 12.6 m


6. 𝜃 is the radian measure of a central angle that intercepts an arc of length 𝑠 in a circle

with a radius 𝑟. If 𝜃 =2𝜋

3 and r = 9, what is the value of s?

a. 18.8

b. 4.3

c. 0.23

d. 56.5

7. A windshield wiper of a car makes an angle of 170°. If the area covered by the blade is 864

square inches, how long is the blade?

a. 1,119,744 inches

b. 36 inches

c. 24 inches

d. 576 inches

8. Given the terminal point of (√2

2,

−√2

2) find tan 𝜃.

a. π

4

b. −π

4

c. -1

d. 1

9. Knowing sec 𝑥 = −5

4 and the terminal point is in the second quadrant find cot 𝜃.

a. −4

5

b. 3

5

c. −4

3

d. −3

4

10. If csc 𝑥 = −13

12 and the terminal point is in the third quadrant, which of the following is NOT true?

a. cos 𝑥 =−5

13

b. tan 𝑥 =12

5

c. sec 𝑥 = −13

5

d. sin 𝑥 =12

13

11. What is the phase shift of 𝑦 =5

3cos(6𝑥 − 2𝜋) + 3?

a. 1

2π

b. π

3

c. 1

3

d. 2𝜋

12. Name the amplitude and vertical shift of 𝑦 = −0.5 cos(3𝑥 + 𝜋) − 3.

a. Amplitude: -0.5, Vertical Shift: -3

b. Amplitude: 0.5, Vertical Shift: -3

c. Amplitude: −𝜋

3, Vertical Shift: 3

d. Amplitude: 𝜋

3, Vertical Shift: -3


13. Which graph represents 𝑦 = −2 cos (3𝑥 −𝜋

3) + 1?

a. c.

b. d.

14. The difference between the maximum of 𝑦 = 2 cos (2 (𝑥 +𝜋

3)) + 1 and 𝑦 = −3 cos(4𝑥 − 𝜋) − 2 is

a. 1

b. 2

c. 3

d. 8

15. (sec 𝑥 + tan 𝑥)(sec 𝑥 − tan 𝑥) =

a. 1 + 2 sec 𝑥 tan 𝑥

b. 1 − sec 𝑥 tan 𝑥

c. 1

d. 1 − sec2 𝑥 sin 𝑥

16. Find the exact value of sin5𝜋

6

a. 1

2

b. −√3

2

c. √3

2

d. √2

2

17. On the interval [0, 2π), if sin 2𝑥 = 0, what is 𝑥?

a. 0

b. π

2

c. 3π

2

d. all of the above

18. If the angle is placed in standard position, its terminal side lies in quadrant II and sin 𝜃 =4

5

What is the value of cos(𝜃 + 3𝜋). (This problem is from the NJ Model Curriculum assessment for

Algebra II Unit 3.) a. −0.8 c. 0.75

b. −0.75 d. 0.8


19.

A mass is attached to a spring, as shown in the figure above. If the mass is pulled down and released, the mass will move up and down for a period of time. The height of the mass above the floor, in inches, can be modeled by the function, f(t), t seconds after the mass is set in motion.

The mass is 4 feet above the floor before it is pulled down. It is pulled 3 inches below the starting

point and makes one full oscillation in 0.2 second. If the spring is at its lowest point at t = 0, which

of the following functions models h ? (This problem is from the NJ Model Curriculum assessment

for Algebra II Unit 3.)

a. 2

48 3cos5

h t t

b. 2

48 3cos5

h t t

c. 48 3cos 10h t t

d. 48 3cos 10h t t

Extended Response

1. Sketch the graph of 𝑦 = −4 sin (2𝑥 −𝜋

3) − 1

2. The water in the bay at Long Beach Island, NJ at a particular pier measures 5 feet deep at

9PM, which is low tide. High tide is reached at 3AM when the gauge reads 12 feet.

a. Which trig function would be the best fit for this model (assuming 9AM is t=0)?

b. Write the equation that models this situation.

c. Determine the amplitude, period, and midline.

d. Predict the water level at midnight.


3. The average daily production, M (in hundreds of gallons), on a dairy farm is modeled by

𝑀 = 19.6 sin (2𝜋𝑑

365+ 12.6) + 45

where d is the day, d=1 is January first.

a. What is the period of the function?

b. What is the average daily production on the last day of the year (d=365)?

c. Using the graph of M(d), what months during the year is production over 5500 gallons a

day?

4. A door has a stained glass window at the top made of panes that are arranged

in a semicircular shape as shown below. The radius of the semicircular shape is 1.5

feet. Its outside edge is trimmed with metal cord. The red sectors are trimmed with

gold cord and the yellow sectors are trimmed with silver cord, as shown in the

diagram below.

a. If all of the sectors are of equal size, how many inches of silver cord will be

needed, and how many inches of gold cord will be needed?

b. What is the total area in square inches of all of the red sectors?


5. A monster truck has tires that are 66 inches in diameter. If a truck rolls a

distance of 100 feet, what is the angle, in radians, that each tire has turned

in rolling that distance?

6. Cal C. was asked to solve the following equation over the interval [0, 2𝜋). During his

calculations he might have made an error. Identify the error and correct his work so that he gets

the right answer.

cos 𝑥 + 1 = sin 𝑥

cos2x + 2 cos x + 1 = 𝑠𝑖𝑛2𝑥

cos2x + 2 cos x + 1 = 1 − 𝑐𝑜𝑠2𝑥

2 cos 𝑥 = 0

cos 𝑥 = 0

π

2,3π

2


Answer Key

For sketches of #1 – 16, see end of key

1. 120°

2. 7𝜋36⁄

3. 5𝜋4⁄

4. 36°

5. 5𝜋6⁄

6. 280°

7. 31𝜋18⁄

8. 257.14°

9. 300°

10. 5𝜋12⁄

11. 10𝜋9⁄

12. 30°

13. 35𝜋36⁄

14. 340°

15. 35𝜋18⁄

16. 231.4°

17. 8𝜋

3, −

4𝜋

3

18. 395°, -325°

19. 585°, -135°

20. 11𝜋

5, −

9𝜋

5

21. 510°, -210°

22. 32𝜋

9, −

4𝜋

9

23. 670°, -50°

24. 24𝜋

7, −

4𝜋

7

25. 11𝜋

3, −

𝜋

3

26. 435°, -285°

27. 560°, -160°

28. 13𝜋

6, −

11𝜋

6

29. 535°, -185°

30. 35𝜋

9, −

𝜋

9

31. 710°, -30°

32. 23𝜋

7, −

5𝜋

7

33. 2 radians

34. 6.3

35. 3 radians

36. 6.4

37. 84.8

38. 0.48

39. 3.3 in2

40. 47.1 in2

41. 8.3 in

42. 97⁄ m

43. -180 °, 4.8 m

44. The car travels 81.7 inches farther

45. 5.2 in.

46. Sally got 33% more (12.6 vs. 9.4 in2)

47. about 37%

48. 89⁄ radians

49. 31.4

50. 4 radians

51. 2.8

52. 4.7

53. 1.4

54. 21.1 in2

55. 99 cm2

56. 3.2 in

57. 116⁄ radians

58. 26𝜋

15 radians, 3.3 in

59. The unicycle goes 50.3 inches farther

60. 9.6 in

61. Billy Bob got 8% more (56.5 vs. 52.5)

62. tan 𝜃 = −2√10

3, 𝜃 = −64.6°

63. cot 𝜃 =5

12, 𝜃 = 247.3°

64. −√5

3

65. −√41

4

66. cos 𝜃 =√15

4, tan 𝜃 = −

√15

15, csc 𝜃 =

−4, sec 𝜃 =4√15

15, cot 𝜃 = −√15

67. cos 𝜃 = −√5

5, sin 𝜃 =

2√5

5, sec 𝜃 = −√5,

csc 𝜃 =√5

2, cot 𝜃 = −

1

2


68. sin 𝜃 =5

8, cos 𝜃 = −

√39

8, tan 𝜃 =

−5√39

39, sec 𝜃 = −

8√39

39, cot 𝜃 = −

√39

5

69. sin 𝜃 = −2√2

3, cos 𝜃 =

1

3, tan 𝜃 = −2√2, csc 𝜃 =

−3√2

4, cot 𝜃 = −

√2

4

70. Quadrant I

71. Quadrant III

72. Quadrant IV

73. Quadrant I

74. − 12⁄

75. − √22

⁄

76. -2

77. √33

⁄

78. -1

79. -1

80. sin4𝜋

3= −

√3

2, cos

4𝜋

3= −

1

2, tan

4𝜋

3= √3

81. sin −𝜋

2= −1, cos −

𝜋

2= 0, tan −

𝜋

2=

𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

82. sin11𝜋

4=

√2

2, cos

11𝜋

4= −

√2

2, tan

11𝜋

4= −1

83. sin 210° = −1

2, cos 210° = −

√3

2, tan 210° =

√3

3

84. sin −315° =√2

2, cos −315° =

√2

2, tan −315° = 1

85. cot 𝜃 = − 724⁄ , and 𝜃 = −73.7°

86. tan 𝜃 = − 7√28

⁄ , and 𝜃 = 128.9°

87. − 8√1515

⁄

88. 34⁄

89. cos 𝜃 = −40

41, tan 𝜃 = −

9

40, csc 𝜃 =

41

9, sec 𝜃 = −

41

40, cot 𝜃 = −

40

9

90. sin 𝜃 =√10

10, cos 𝜃 = −

3√10

10, tan 𝜃 =

−1

3, csc 𝜃 = √10, sec 𝜃 = −

√`10

3

91. sin 𝜃 = −4

5, tan 𝜃 =

4

3, csc 𝜃 = −

5

4, sec 𝜃 =

−5

3, cot 𝜃 =

3

4

92. cos 𝜃 = −0.7, tan 𝜃 = −1, csc 𝜃 =10

7, sec 𝜃 = −

10

7, cot −1

93. Quadrant II

94. Quadrant IV

95. Quadrant I

96. Quadrant IV

97. 12⁄

98. √22

⁄

99. -2

100. −√33

⁄

101. 1

102. 1

103. sin8𝜋

3=

√3

2, cos

8𝜋

3= −

1

2, tan

8𝜋

3= −√3

104. sin5𝜋

4= −

√2

2, cos

5𝜋

4= −

√2

2, tan

5𝜋

4= 1

105. sin −7𝜋

6=

1

2, cos −

7𝜋

6= −

√3

2, tan −

7𝜋

6=

−√3

3

106. sin 690° = −1

2, cos 690° =

√3

2, tan 690° = −

√3

3

107. sin −240° =√3

2, cos −240° =

−1

2, tan −240° = − √3

108. a. 2, b. 2, c. 3, d. 1, e. 1, f. 2, g. 4

109. 𝑎. 2𝜋, 𝑏. 𝜋, 𝑐. 4𝜋, 𝑑. 2𝜋, 𝑒. 2𝜋, 𝑓. 𝜋, 𝑔. 4𝜋

110. 𝑎. 0, 𝑏. 0, 𝑐. 0, 𝑑.𝜋

3, 𝑒. −

𝜋

4, 𝑓.

𝜋

6, 𝑔. −2𝜋

111. a. 0, b. 0, c. 1, d. 0, e. 0, f. 0, g. 1

112. a.


b.

c.

d.

e.

f.

g. 113. No they are not the same, they are

reflections of each other. For example,

when x = 0, cos x = 1, and sin (x - 𝜋

2) = -1.

114. A: 2, P: 𝜋, VS: -1, y = 2 cos (2x) −1 (one

possible answer)

115. A: 1, P: 𝜋, VS: 3, y = sin (2x) + 3

116. A: 4, P: 𝜋, VS: 0, y = 4 sin (2𝑥)

117. A: 0.5, P: 2 𝜋, VS: 0, y = -0.5 cos x

118. A: 2, P: 2𝜋, PS:− 𝜋3⁄ , VS: 1

119. A: 3, P: 𝜋 2⁄ , PS: 𝜋 4⁄ , VS: -2


120. A: 1, P: 3𝜋, PS: − 𝜋6⁄ , VS: 3

121. A: 1, P: 2𝜋3⁄ , PS: 2𝜋

3⁄ , VS: -1

122. A: 2 3⁄ , P: 𝜋 2⁄ , PS: 𝜋 2⁄ , VS: 2

123. 1440⁄

124. 𝑦 = −5 cos2𝜋

11𝑥

125. a. 3, b. 2, c. 1, d. 1, e. 2, f. 4, g. 2

126. a. 2 𝜋, b. 𝜋, c. 12 𝜋, d. 2 𝜋, e. 2 𝜋, f. 2 𝜋, g.

2 𝜋

127. a. 0, b. 0, c. 0, d. -2𝜋

3, e. -

𝜋

4, f.

𝜋

3, g. -3 𝜋

128. a. 0, b. 0, c. 0, d. 0, e. 0, f. -2, g. 5

129. a.

b.

c.

d.

e.

f.

g.


130. A: 4, P: 4𝜋, PS: 𝜋 3⁄ , VS: 2

131. A: 2, P: 𝜋 2⁄ , PS: 3𝜋4⁄ , VS: -3

132. A; 2, P: 8𝜋, PS: − 𝜋2⁄ , VS: 1

133. A: 1, P: 𝜋 3⁄ , PS: 𝜋 3⁄ , VS: -1

134. A: 3 2⁄ , P: 𝜋 2⁄ , PS: 3𝜋4⁄ , VS: -2

135. 1

550,

1

660

136. 𝑦 = 3.5 cos (𝜋

9𝑥)

137. sec x

138. 1

139. cos2 x

140. 1

141. cos2 x

142. 1 – 2cos 𝑥 sin 𝑥

143. csc2 x

144. 1 – sin 𝑥

145. sec 𝑥

146. 1- sin2 𝑥 = cos2 𝑥

147. sin 𝑥

cos 𝑥∙cos 𝑥

sin 𝑥

sec 𝑥=

1

sec 𝑥= cos 𝑥

148. sin2 𝑥 ∙ sec2 𝑥 = sin2 𝑥 ∙

1

cos2 𝑥=

sin2 𝑥

cos2 𝑥= tan2 𝑥

149. sec 𝑥−tan 𝑥

(sec 𝑥+tan 𝑥)(sec 𝑥−tan 𝑥)+

sec 𝑥+tan 𝑥

(sec 𝑥+tan 𝑥)(sec 𝑥−tan 𝑥)=

2 sec 𝑥

sec2 𝑥 − tan2 𝑥=

2 sec 𝑥

1

150. Sec2x + Csc2x

151. 2Secx

152. 0

153. CosxCotx


154. Cos2x + 1

155. 1

156. Sin2x

157. 1

158. 2

159. (1 − 𝑠𝑖𝑛2𝑥) − 𝑠𝑖𝑛2𝑥 =

1 − 2𝑠𝑖𝑛2𝑥

160. 𝑠𝑖𝑛𝑥

𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑥

1

𝑠𝑖𝑛𝑥= 1

161. (1 +cos 𝑥

sin 𝑥) sin 𝑥 = sin 𝑥 + cos 𝑥

162. (𝑐𝑜𝑠𝑥) (1

𝑠𝑖𝑛𝑥) (

𝑠𝑖𝑛𝑥

𝑐𝑜𝑠𝑥) =

(𝑐𝑜𝑠𝑥

𝑠𝑖𝑛𝑥) (

𝑠𝑖𝑛𝑥

𝑐𝑜𝑠𝑥) =

1

MC1. C

MC2. D

MC3. C, D

MC4. A

MC5. C

MC6. A

MC7. C

MC8. C

MC9. C

MC10. D

MC11. B

MC12. B

MC13. C

MC14. B

MC15. C

MC16. A

MC17. D

MC18. D

MC19. C

ER1.

ER2A. cosine

ER2B. 𝑦 = −3.5 cos (𝜋

6𝑡) + 8.5

ER2C. amplitude = 3.5, period = 12, midline = 8.5 ft

ER2D. 8.5 feet

ER3A. 365 days

ER3B. 4,500 Gallons

ER4A. 22.6 inches, 33.9 inches

ER4B. 305.4 square inches

ER5. 3 radians

ER6. 2𝑐𝑜𝑠2𝑥 + 2𝑐𝑜𝑠𝑥 = 0

𝑐𝑜𝑠2𝑥 + 𝑐𝑜𝑠𝑥 = 0

𝜋

2,𝜋,

3𝜋

2


1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

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