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Algebra / Geometry III:
Unit 8- Periodic Trig
SUCCESS CRITERIA:
1. Be able to find an additional positive and negative co-terminal angle in degrees and radians.
2. Be able to use the unit circle to find the exact value of special angles.
3. Be able to graph trig functions and their transformations finding key features.
4. Be able to transform trig expressions and solve equations using trig identities.
INSTRUCTOR: Craig Sherman
Hidden Lake High School
Westminster Public Schools
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EMPOWER Recorded TARGET
SCALE THEME
MA.11.F.05.04
Extend the Domain of Trigonometric Functions Using the Unit Circle
MA.11.F.06.04
Periodic Phenomena
MA.11.F.07.04
Trigonometric Identities
MA.11.G.01.04
Apply Trigonometry
VOCABULARY
o
o
o
o
radian measure
terminal side
co-terminal side
unit circle
o
o
o
o
cotangent
secant
cosecant
amplitude
o
o
o
o
frequency
phase shift
period
trig idenity
PROFICIENCY SCALE:
SCORE
REQUIREMENTS
4.0
In addition to exhibiting Score 3.0 performance, in-depth inferences and applications that go BEYOND what
was taught in class.
Score 4.0 does not equate to more work but rather a higher level of performance.
3.5
3.0
In addition to Score 3.0 performance, in-depth inferences and applications with partial success.
The learner exhibits no major errors or omissions regarding any of the information and processes (simple or
complex) that were explicitly taught.
o Be able to find an additional positive and negative co-terminal angle in degrees and radians
AND
o Be able to use the unit circle to find the exact value of special angles, AND
o Graph polynomial equations and identify its key features, AND
o Be able to graph trig functions and their transformations finding key features, AND
o Be able to transform trig expressions and solve equations using trig identities.
2.0
Can do one or more of the following skills / concepts:
There are no major errors or omissions regarding the simpler details and processes as the learner…
o Convert angle degrees into radians, OR
o Convert radian measures to degrees, OR
o Graph periodic trig functions, OR
o Identify key features of periodic functions, OR
o Graph periodic transformations, OR
o Identify key features of periodic transformations, OR
o Simplify trig expressions using the trig identities, OR
o Use trig identities to solve trig proofs.
1.0
Know and use the vocabulary
Identify the Basic Elements
With help, a partial understanding of some of the simpler details and process
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Converting Degrees and Radians
INSTRUCTION 1: KHAN ACADEMY
INSTRUCTION 2: SOPHIA
Class Work
Convert the following degree measures to radians and radian measures to degrees. Sketch each angle.
2𝜋
5. 150°
1.
3
14𝜋
9
2. 35°
6.
3. 225°
7. 310°
4.
π
5
8.
10π
7
Homework
Convert the following degree measures to radians and radian measures to degrees. Sketch each angle.
π
5𝜋
12. 6
9.
3
10. 75°
13. 175°
11. 200°
14.
17𝜋
9
15. 350°
16.
9π
7
Co-terminal Angles
INSTRUCTION 1: SOPHIA
Classwork
Name one positive angle and one negative angle that is co-terminal with the given angle.
2𝜋
21. 150°
17.
3
14𝜋
9
18. 35°
22.
19. 225°
23. 310°
π
20. 5
24.
10π
7
Homework
Name one positive angle and one negative angle that is co-terminal with the given angle.
5𝜋
29. 175°
25.
3
17𝜋
9
26. 75°
30.
27. 200°
31. 350°
π
28. 6
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32.
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9π
7
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INSTRUCTION 1: KHAN ACADEMY
INSTRUCTION 2: SOPHIA
UNIT CIRCLE
TRIG VALUES
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
( _______ ,________ )
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( _______ ,________ )
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Class Work
3 −2√10
)
7
7
33. Given the terminal point ( ,
−5 −12
)
13
34. Given the terminal point ( 13 ,
find tanθ and 𝜃.
find cot 𝜃 and 𝜃.
2
35. Given cos 𝜃 = 3 and the terminal point in the fourth quadrant, find sin 𝜃.
4
36. Given cot 𝜃 = 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.
1
37. sin 𝜃 = − 4 and the terminal point is in the fourth quadrant.
38. tan 𝜃 = −2 and the terminal point is in the second quadrant.
8
5
39. csc 𝜃 = and the terminal point is in the second quadrant.
40. 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. cos
4π
3
76. sec
2π
3
78. cot
15π
4
75. sin
7π
4
77. tan
-5π
6
79. csc
-9π
2
Find the exact value of the sine, cosine and tangent of the given angle.
80.
4𝜋
3
𝜋
81. – 2
82.
11𝜋
4
83. 210°
84. -315°
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Homework
7 −24
,
)
25 25
find cotθ and 𝜃.
−4√2 7
, 9)
9
find tanθ and 𝜃.
85. Given the terminal point (
86. Given the terminal point (
7
87. Given sin 𝜃= 8 and the terminal point in the second quadrant, find sec 𝜃.
5
88. Given csc 𝜃 = −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.
9
89. sin 𝜃 = 41 and the terminal point is in the second quadrant.
90. cot 𝜃 = −3 and the terminal point is in the second quadrant.
3
5
91. cos 𝜃 = − 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.
4π
3
97. cos
5π
3
99. sec
98. sin
3π
4
100. tan
−7π
6
101. cot
13π
4
102. csc
−11𝜋
2
Find the exact value of the sine, cosine and tangent of the given angle.
7𝜋
6
103.
8𝜋
3
105.
−
104.
5𝜋
4
106.
690°
107.
-240°
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Graphing
INSTRUCTION 1 : KHAN ACADEMY
INSTRUCTION 2 SOPHIA
Function Transformations
± a function ( ± b X ± c ) ± d
ACTIONS
DIRECTION
Reflection ( - )
Vertical (outside)
Stretch/Shrink ( a &
b)
Horizontal
(inside)
Phase Shift ( ± c & d)
INSTRUCTION 1 : KHAN ACADEMY
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INSTRUCTION 2 SOPHIA
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Classwork
Use the functions below to answer questions 108 – 111.
a. 𝑦 = 2 cos 𝑥
b. 𝑦 = −2 sin 2𝑥
𝑥
c. 𝑦 = −3 sin 2 + 1
𝜋
4
e. 𝑦 = sin (𝑥 + )
𝜋
d. 𝑦 = cos (𝑥 − 3 )
f.
𝜋
3
𝑦 = 2 cos (2𝑥 − )
g. 𝑦 = −4 sin(0.5𝑥 + 𝜋) + 1
108.
Find the amplitude of each function.
110.
Find the phase shift of each function.
109.
Find the period 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 − )? Justify your answer
𝜋
2
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
check with a graphing calculator.
𝜋
2
𝜋
121. 𝑦 = −1 cos(3𝑥 − 2𝜋) − 1
118. 𝑦 = 2 cos (𝑥 + ) + 1
120. 𝑦 = sin ( (𝑥 + )) + 3
3
119.
3
6
122.
𝑦 = −3 cos(4𝑥 − 𝜋) − 2
2
𝑦 = 3 cos(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.
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Homework
Use the functions below to answer questions 125 – 128.
a. 𝑦 = −3 cos 𝑥
b. 𝑦 = −2 sin 2𝑥
𝑥
c. 𝑦 = − sin 6
d. 𝑦 = cos (𝑥 +
𝜋
e. 𝑦 = −2 sin (𝑥 + 4 )
f.
2𝜋
)
3
𝜋
𝑦 = 4 cos (𝑥 − 3 ) − 2
g. 𝑦 = −2 sin(𝑥 + 3𝜋) + 5
125. Find the amplitude of each function.
127. Find the phase shift of each function.
126. Find the period 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 check
with a graphing calculator.
1
2
𝜋
3
130. 𝑦 = −4 cos ( (𝑥 − )) + 2
131. 𝑦 = −2 cos(4𝑥 − 3𝜋) − 3
1
4
𝜋
2
132. 𝑦 = 2 sin ( (𝑥 + )) + 1
133. 𝑦 = −1 cos(6𝑥 − 2𝜋) − 1
3
2
134. 𝑦 = cos(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.
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Trigonometric Identities
INSTRUCTION 1 : KHAN ACADEMY
Class Work
Simplify the expression
137. csc 𝑥 tan 𝑥
INSTRUCTION 2 SOPHIA
139. sin x (csc x − sin x)
138. cot 𝑥 sec 𝑥 sin 𝑥
140. (1 + cot 2 x)(1 − cos 2 x)
141. 1 − tan2 x ÷ sec 2 𝑥
142. (sin x − cos x)2
143.
cos 𝑥
cot2 x
1−sin2x
145. sin 𝑥 tan 𝑥 + cos 𝑥
144. sec 𝑥+tan 𝑥
Verify the Identity
146. (1 − sin 𝑥)(1 + sin 𝑥) = cos 2 x
147.
tan 𝑥 cot 𝑥
sec 𝑥
= cos 𝑥
148. (1 − cos2 x)(1 + tan2 x) = tan2 x
149.
1
sec x+tan x
+
1
sec x−tan x
= 2 sec x
Homework
Simplify the expression
150. (tan x + cot x )2
1
151.
1
153. sin 𝑥 − csc 𝑥
sin2 x
cos2 x
155. tan2 x + cot2 x
157.
cos x
sec x
+
sin x
csc x
Verify the Identity
159. 𝑐𝑜𝑠 2 𝑥 − 𝑠𝑖𝑛2 𝑥 = 1 − 2𝑠𝑖𝑛2 𝑥
161.
1+cot x
csc x
= sin x + cos x
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1−sin x
cos x
+ 1−sin x
cos x
152.
154.
1+sec2 x
1+tan2 x
156.
𝑡𝑎𝑛2 𝑥
1+𝑡𝑎𝑛2 𝑥
158.
1+sec2 x
cos2 x
+ 2
1+tan2 x
cot x
cos x−cos y
sin x+sin y
sin x−sin y
+ cos x+cos y
160. tan 𝑥 cos 𝑥 csc 𝑥 = 1
162.
cos x csc x
cot x
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PERIODIC TRIG
UNIT REVIEW
Multiple Choice
1. How many degrees is
a.
b.
c.
d.
4π
9
?
160°
110°
80°
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°
c. -190°
b. 190°
d. 530°
4. Which is larger and by how much: an angle of 258°, or an angle of
6
a. 258° by °
c.
b. 258° by radian
d.
7
6
7
5. The central angle of a circle has a measure of
5𝜋
4
10𝜋
radians?
7
10𝜋
7
10𝜋
7
1
radians by °
7
6
radians by °
7
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
c. 1.3 m
b. 2.0 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
c. 0.23
b. 4.3
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
c. 24 inches
b. 36 inches
d. 576 inches
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8. Given the terminal point of (
a.
√2 −√2
,
2
2
) find tan 𝜃.
π
4
b. −
π
4
c. -1
d. 1
5
9. Knowing sec 𝑥 = − and the terminal point is in the second quadrant find cot 𝜃.
4
a.
b.
−4
c.
5
3
d.
5
10. If csc 𝑥 = −
a. cos 𝑥 =
b. tan 𝑥 =
13
12
−5
−4
3
−3
4
and the terminal point is in the third quadrant, which of the following is NOT true?
c.
13
12
sec 𝑥 = −
d. sin 𝑥 =
5
13
5
12
13
5
11. What is the phase shift of 𝑦 = cos(6𝑥 − 2𝜋) + 3?
3
a.
b.
1
c.
2π
π
1
3
d. 2𝜋
3
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
𝜋
c. Amplitude: − , Vertical Shift: 3
d. Amplitude:
𝜋
3
3
, Vertical Shift: -3
𝜋
13. Which graph represents 𝑦 = −2 cos (3𝑥 − ) + 1?
3
a.
c.
b.
d.
𝜋
14. The difference between the maximum of 𝑦 = 2 cos (2 (𝑥 + )) + 1 and 𝑦 = −3 cos(4𝑥 − 𝜋) − 2 is
3
a. 1
b. 2
15. (sec 𝑥 + tan 𝑥)(sec 𝑥 − tan 𝑥) =
a. 1 + 2 sec 𝑥 tan 𝑥
b. 1 − sec 𝑥 tan 𝑥
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c. 3
d. 8
c. 1
d. 1 − sec 2 𝑥 sin 𝑥
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16. Find the exact value of sin
a.
b.
5𝜋
6
1
c.
2
−√3
d.
2
√3
2
√2
2
17. On the interval [0, 2π), if sin 2𝑥 = 0, what is 𝑥?
a. 0
π
b.
c.
2
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 = What is the value of
5
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
4
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 
h t   48  3cos 
t
 5 
b.
 2
h t   48  3cos 
 5
c.
h t   48  3cos 10 t 
d.
h t   48  3cos 10 t 

t

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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.
3.
Predict the water level at midnight.
The average daily production, M (in hundreds of gallons), on a dairy farm is modeled by
2𝜋𝑑
𝑀 = 19.6 sin (
+ 12.6) + 45
365
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
cos x + 2 cos x + 1 = 𝑠𝑖𝑛2 𝑥
cos 2 x + 2 cos x + 1 = 1 − 𝑐𝑜𝑠 2 𝑥
2 cos 𝑥 = 0
cos 𝑥 = 0
π 3π
,
2 2
2
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