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Converting Degrees and Radians – Class WorkConvert the following degree measures to radians and radian measures to degrees. Sketch each angle.

1.2π3

2. 35 °

3. 225 °

4.π5

5. 150 °

6.14π

9

7. 310 °

10π7 Converting Degrees and Radians – Homework

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

8.5π3

9. 75 °

10. 200 °

11.π6

12. 175 °

13.17π

9

14. 350 °

15.9π7

Co-terminal Angles – ClassworkName one positive angle and one negative angle that is co-terminal with the given angle.

17.2π3

18. 35 °

19. 225 °

20.π5

21. 150 °

22.14π

9

23. 310 °

Alg 2-Trig Functions ~1~ NJCTL.org

10π7


Co-terminal Angles – HomeworkName one positive angle and one negative angle that is co-terminal with the given angle.

25.5π3

26. 75 °

27. 200 °

28.π6

29. 175 °

30.17π

9

31. 350 °

32.9π7

Arc Length and Sector Area - ClassworkRound 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 andr=5 , findθ.

34. If θ=π3andr=6 , find s .

35. If s=5.4∧r=1.8 , find θ .

36. If s=15∧θ=3 π4, find r .

37. If r=9∧θ=3 π , find s .

38. If θ=6π∧s=9 , find r .

39. Find the area of a sector with radius 5 inches and central angle θ= π12 .

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=5 π12 .

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 - HomeworkFor 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 , findθ.

49. If θ=5 π3

and r=6 ,find s .

50. If s= .001∧r=.00025 , find θ .

51. If s=20∧θ=9π4, find r .

52. If r=1.5∧θ=π , find s .

53. If θ=4 π∧s=18 , find r .

54. Find the area of a sector with radius 11 inches and central angle θ=π9 .


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=3 π4 .

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 38of an 18-inch pizza pie and Sally Sue got

49 of a 16-inch pie. Who got

more pizza and by what percent?

Unit Circle – Class Work

62. Given the terminal point ( 37,−2√10

7 ) find tanθ and θ.

63. Given the terminal point (−513,−12

13 ) find cot θ andθ .


64. Given cos θ =23and the terminal point in the fourth quadrant, find sin θ.

65. Given cot θ =45 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=−14 and the terminal point is in the fourth quadrant.

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

68. csc θ=85 and the terminal point is in the second quadrant.

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

State the quadrant in which θ lies:

70. sin θ>0 ,cosθ>071. sin θ<0 , tanθ>072. csc θ<0 , secθ>073. sin θ>0 , cot θ>0

Find the exact value of the given expression.

74. cos 4π3

75. sin 7π4

76. sec 2π3

77. tan-5π6


78. cot 15π4

csc -9π2 Find the exact value of the sine, cosine and tangent of the given angle.

80.4 π3

81. – π2

82.11π

4

83. 210°

84. -315°

Unit Circle – Homework

85. Given the terminal point ( 725,−24

25 ) find cotθ∧θ.

86. Given the terminal point (−4 √29

, 79 ) find tanθ∧θ.

87. Given sin θ=78and the terminal point in the second quadrant, find secθ.

88. Given cscθ = 5

−4 and the terminal point in the third quadrant find cotθ.

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

89. sin θ= 941 and the terminal point is in the second quadrant.


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

91. cosθ=−35 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 ,cosθ<094. sin θ<0 , tanθ<095. csc θ>0 , secθ>096. sin θ<0 , cot θ<0

Find the exact value of the given expression.

97. cos 5π3

98. sin 3π4

99. sec 4 π3

100. tan −7 π6

101. cot 13π4

102. csc−11π2

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

103.8π3


104.5π4

105.−7π

6

106. 690°

107. -240°

Graphing ClassworkUse the functions below to answer questions 108 – 111.

a. y=2cos x b. y=−2sin 2 x

c. y=−3sin x2+1 d. y=cos(x− π3 )

e. y=sin(x+ π4 ) f. y=2cos (2 x−π3 )g. y=−4sin (0.5 x+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−π2 )? 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=2cos (x+ π3 )+1

119. y=−3cos (4 x−π )−2

120. y=sin( 23 ( x+ π6 ))+3

121. y=−1cos (3 x−2 π )−1

122. y=23

cos (4 x−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(880 πx), 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 – HomeworkUse the functions below to answer questions 124 – 127.

a. b. y=−2sin 2 x

c. y=−sin x6 d. y=cos(x+ 2π

3 )e. y=−2sin (x+ π4 ) f. y=4 cos (x−π3 )−2

g. y=−2sin ( x+3π )+5

h.

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=−4cos (12 ( x−π3 ))+2

131. y=−2cos (4 x−3 π )−3

132. y=2sin( 14 (x+ π2 ))+1

133. y=−1cos (6x−2π )−1

134. y=32

cos (4 x−3π )−2

135. The musical notes C# (C sharp) and E can be modeled by the sine wavesy=sin(1100 πx), and y=sin (1320 πx ) 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 WorkSimplify the expression

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


140. (1+cot2 x ) (1−cos2 x ) 141. 1−ta n2 x ÷ s ec2 x

142. (sin x−cos x )2 143. cot2 x1−sin2 x

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 xcot xsec x

=cos x

148. (1−cos2 x ) (1+ta n2x )=tan2 x 149. 1

sec x+ tan x+ 1sec 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 ysin x+sin y

+ sin x−sin ycos x+cos y

153. 1

sin x− 1csc x 154.

1+se c2 x1+ta n2x

155. sin 2 xtan2 x

+ co s2 x

co t2 x156. ta n

2 x1+ tan2 x

157. cos xsec x

+ sin xcsc x 158.

1+se c2 x1+ta n2x

+ co s2 x

co t 2 x

Verify the Identity159. cos2x−si n2x=1−2 si n2 x 160. tan x cos x csc x=1

161. 1+cot xcsc x

=sin x+cos x 162. cos x csc x

cot x=1


Unit ReviewMultiple Choice

1. How many degrees is 4 π9?

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

7radians?

a. 258° by 67°

b. 258° by 67 radian

c.10π

7 radians by

17°

d.10π

7 radians by

67°


5. The central angle of a circle has a measure of 5π4 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 mb. 2.0 mc. 1.3 md. 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=2 π3 and r = 9, what is the value of s?

a. 18.8b. 4.3c. 0.23d. 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 inchesb. 36 inchesc. 24 inchesd. 576 inches

8. Given the terminal point of (√22,−√2

2 ) find tanθ .

a.π4

b.−π4

c. -1d. 1

9. Knowing sec x=−54

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

a.−45

b.35

c.−43

d.−34

10. If csc x=−1312 and the terminal point is in the third quadrant, which of the following is NOT true?

a. cos x=−513


b. tan x=125

c. sec x=−135

d. sin x=1213

11. What is the phase shift of y=53

cos (6 x−2π )+3?

a.1

2π

b.π3

c.13

d. 2π12. Name the amplitude and vertical shift of y=−0.5 cos (3 x+π )−3 .

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

c. Amplitude: −π

3 , Vertical Shift: 3

d. Amplitude: π3 , Vertical Shift: -3

13. Which graph represents y=−2cos (3 x−π3 )+1?

a. c.

b. d.


14. The difference between the maximum of y=2cos (2(x+ π3 ))+1 and y=−3 cos (4 x−π )−2 is

a. 1b. 2c. 3d. 8

15. ( sec x+ tan x ) (sec x−tan x )=¿a. 1+2 sec x tan xb. 1−sec x tan xc. 1d. 1−se c2 x sin x

16. Find the exact value of sin 5π6

a.12

b. −√32

c. √32

d. √22

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

b.π2

c.3π2

d. all of the above

18. If the angle is placed in standard position, its terminal side lies in quadrant II and sin θ=45

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

for Algebra II Unit 3.)a. −0.8 c. 0.75b. −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.

b.

c.

d.

Extended Response

1. Sketch the graph of y=−4 sin (2 x−π3 )−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( 2 πd365

+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 ¿cos2 x+2 cos x+1=sin2 x

cos2 x+2 cos x+1=1−cos2 x2cos x=0cos x=0π2, 3 π

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

11.10π

912. 30°

13.35π36

14. 340°

15.35π18

16. 231.4°

17.8π3,−4 π

318. 395°, -325°19. 585°, -135°

20.11π

5,−9 π

521. 510°, -210°

22.32π

9,−4 π

923. 670°, -50°

24.24 π

7,− 4π

7

25.11π

3,− π

326. 435°, -285°27. 560°, -160°

28.13π

6,−11π

629. 535°, -185°

30.35π

9,−π

931. 710°, -30°

32.23π

7,−5π

733. 2 radians34. 6.335. 3 radians36. 6.437. 84.838. 0.4839. 3.3 in2

40. 47.1 in2

41. 8.3 in

42.97 m

43. -180 °, 4.8 m44. The car travels 81.7 inches farther45. 5.2 in.46. Sally got 33% more (12.6 vs. 9.4 in2)47. about 37%

48.89 radians

49. 31.450. 4 radians51. 2.852. 4.753. 1.454. 21.1 in2

55. 99 cm2 56. 3.2 in

57.116 radians

58.26π15 radians, 3.3 in

59. The unicycle goes 50.3 inches farther60. 9.6 in


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

62. tan θ=−2√103

, θ=−64.6 °

63. cot θ= 512,θ=247.3 °

64. −√53

65. −√414

66.

cosθ=√154, tan θ=−√15

15, cscθ=−4 , sec θ=4 √15

15,cot θ=−√15

67. cos

θ=−√55, sinθ=2√5

5, secθ=−√5,

csc θ=√52,cot θ=−1

268.

sin θ=58,cosθ=−√39

8, tan θ=−5√39

39, sec θ=−8√39

39,cotθ=−√39

569.

sin θ=−2√23

,cosθ=13, tan θ=−2√2 , csc θ=−3√2

4,cot θ=−√2

470. Quadrant I71. Quadrant III72. Quadrant IV73. Quadrant I

74.−12

75. −√22

76. -2

77. √33

78. -179. -180.

sin 4 π3

=−√32,cos 4 π

3=−1

2, tan 4 π

3=√3

81.

sin −π2

=−1, cos −π2

=0 , tan −π2

=undefined

82.

sin 11π4

=√22,cos 11π

4=−√2

2, tan 11 π

4=−1

83.

sin 210 °=−12,cos210 °=−√3

2, tan210 °=√3

384.

sin−315 °=√22,cos−315 °=√2

2, tan−315 °=1

85. cot θ=¿− 724

, andθ=−73.7 ° ¿

86. tanθ=¿−7 √28

¿, and θ = 128.9°

87. −8√1515

88.34

89.

cosθ=−4041

, tan θ=−940, cscθ=41

9, secθ=−41

40,cotθ=−40

990.

sin θ=√1010

,cosθ=−3√1010

, tan θ=−13, cscθ=√10 , sec θ=−√ 10

391.

sin θ=−45, tan θ= 4

3,csc θ=−5

4, sec θ=−5

3,cot θ=3

492.

cosθ=−0.7 , tanθ=−1 , csc θ=107, secθ=−10

7,cot−1

93. Quadrant II94. Quadrant IV95. Quadrant I96. Quadrant IV

97.12

98. √22

99. -2


100. −√33

101. 1102. 1103.

sin 8π3

=√32,cos 8 π

3=−1

2, tan 8π

3=−√3

104.

sin 5π4

=−√22,cos 5π

4=−√2

2, tan 5π

4=1

105.

sin −7π6

=12,cos −7 π

6=−√3

2, tan −7π

6=−¿ √3

3¿

106.

sin 690 °=−12,cos690 °=√3

2, tan690 °=¿−√3

3¿

107.

sin−240 °=√32,cos−240 °=−1

2, tan−240°=¿−√3¿

108. a. 2, b. 2, c. 3, d. 1, e. 1, f. 2, g. 4109.a .2π ,b .π , c .4 π ,d .2π , e .2π , f . π , g .4 π110.

a .0 , b .0 , c .0 , d . π3, 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 - π2 ) = -

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: 2π, VS: 0, y = -0.5 cos x

118. A: 2, P:2π , PS:−π3 , VS: 1

119. A: 3, P: π2 , PS:

π4 , VS: -2

120. A: 1, P: 3π, PS: −π6 , VS: 3

121. A: 1, P: 2π3 , PS:

2π3 , VS: -1

122. A: 23 , P:

π2 , PS:

π2 , VS: 2

123.1

440

124. y=−5cos 2 π11x

125. a. 3, b. 2, c. 1, d. 1, e. 2, f. 4, g. 2126. a. 2π , b. π , c. 12π, d. 2π, e. 2π, f. 2π, g.

2π

127. a. 0, b. 0, c. 0, d. -2π3 , e. -

π4 , f.

π3 , g. -3

π128. 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: π3 , VS: 2

131. A: 2, P: π2 , PS:

3π4 , VS: -3

132. A; 2, P: 8π, PS: −π2 , VS: 1


133. A: 1, P: π3 , PS:

π3 , VS: -1

134. A: 32 , P:

π2 , PS:

3π4 , VS: -2

135. 1550

, 1660

136. y=3.5 cos ( π9 x)137. sec x138. 1139. cos2 x140. 1141. cos2 x142. 1 – 2 cos x sin x 143. csc2 x144. 1 – sin x145. sec x146. 1-sin2 x= cos2 x

147. sin xcos x

∙ cos xsin x

sec x= 1sec x

=cos x

148.sin2 x ∙ sec2 x=¿ sin2 x ∙ 1

cos2 x= sin2 x

cos2 x=tan2x ¿

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 xsec2 x−tan 2 x

=2 sec x1

150. Sec2x + Csc2x

151. 2Secx

152. 0

153. CosxCotx

154. Cos2x + 1

155. 1

156. Sin2x

157. 1

158. 2

159. (1−sin2 x )−sin2 x=¿

1−2 sin2 x

160.sinxcosx

cosx 1sinx

=1

161. (1+ cos xsin x )sin x=sin x+cos x

162. (cosx )( 1sinx )( sinxcosx )=¿

( cosxsinx )( sinxcosx )=¿

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( π6 t )+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. 2 cos2 x+2cosx=0

cos2 x+cosx=0

π2, π , 3 π

2


1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.


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