**Algebra / Geometry III: **Unit 8-Periodic Trig

**SUCCESS CRITERIA:**

- Be able to find an additional positive and negative co-terminal angle in degrees and radians.

- Be able to use the unit circle to find the exact value of special angles.

- Be able to graph trig functions and their transformations finding key features.

- Be able to transform trig expressions and solve equations using trig identities.

INSTRUCTOR: Craig Sherman

**Hidden Lake High School**

**Westminster Public Schools**

EMPOWER Recorded TARGETSCALE THEME

MA.11.F.05.04Extend the Domain of Trigonometric Functions Using the Unit Circle

MA.11.F.06.04Periodic Phenomena

MA.11.F.07.04Trigonometric Identities

MA.11.G.01.04Apply Trigonometry

VOCABULARY

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- radian measure
- terminal side
- co-terminal side
- unit circle
- cotangent
- secant
- cosecant
- amplitude
- frequency
- phase shift
- period
- trig idenity

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**PROFICIENCY SCALE:**

**SCOREREQUIREMENTS**

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 In addition to Score 3.0 performance, in-depth inferences and applications with partial success.

3.0The learner exhibits no major errors or omissions regarding any of the information and processes (simple or complex) that were explicitly taught.

oBe able to find an additional positive and negative co-terminal angle in degrees and radiansAND

oBe able to use the unit circle to find the exact value of special angles, AND

oGraph polynomial equations and identify its key features, AND

oBe able to graph trig functions and their transformations finding key features, AND

oBe able to transform trig expressions and solve equations using trig identities.

2.0Can 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…

oConvert angle degrees into radians, OR

oConvert radian measures to degrees, OR

oGraph periodic trig functions, OR

oIdentify key features of periodic functions,OR

oGraph periodic transformations, OR

oIdentify key features of periodic transformations, OR

oSimplify trig expressions using the trig identities, OR

oUse 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

**Converting Degrees and Radians**

INSTRUCTION 1: KHAN ACADEMYINSTRUCTION 2: SOPHIA

Class Work

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

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Homework

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

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**Co-terminal Angles**

INSTRUCTION 1: SOPHIA

Classwork

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

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- 1

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Homework

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

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INSTRUCTION 1: KHAN ACADEMYINSTRUCTION 2: SOPHIA

UNIT CIRCLETRIG VALUES

( ______,______)

( ______,______) / ( ______,______)

( ______,______)

( ______,______)

/ ( ______,______)

( ______,______) / ( ______,______)

( ______,______) / ( ______,______)

( ______,______) / ( ______,______)

( ______,______) / ( ______,______)

( ______,______) / ( ______,______)

( ______,______) / ( ______,______)

( ______,______)

( ______,______) / ( ______,______)

Class Work

- Given the terminal point and .
- Given the terminal point
- Given cos=and the terminal point in the fourth quadrant, find sin.
- Given cot =and the terminal point in the third quadrant, find sec .

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

- and the terminal point is in the fourth quadrant.
- and the terminal point is in the second quadrant.
- and the terminal point is in the second quadrant.
- and the terminal point is in the fourth quadrant.

State the quadrant in which lies:

Find the exact value of the given expression.

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

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- 210
- -315

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Homework

- Given the terminal point .
- Given the terminal point .
- Given sin =and the terminal point in the second quadrant, find sec.
- Given csc = and the terminal point in the third quadrant find cot.

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

- and the terminal point is in the second quadrant.
- and the terminal point is in the second quadrant.
- and the terminal point is in the third quadrant.
- and the terminal point is in the second quadrant.

State the quadrant in which lies:

Find the exact value of the given expression.

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

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- 690
- -240

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Graphing

INSTRUCTION 1 :KHAN ACADEMYINSTRUCTION 2 SOPHIA

Function Transformations±a function ( ± b X ± c ) ± d

ACTIONS / DIRECTION

Reflection ( - ) / Vertical (outside)

Stretch/Shrink (ab) / Horizontal (inside)

Phase Shift ( ± cd)

INSTRUCTION 1 :KHAN ACADEMYINSTRUCTION 2 SOPHIA

Classwork

Use the functions below to answer questions 108 – 111.

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- Find the amplitude of each function.
- Find the period of each function.
- Find the phase shift of each function.
- Find the vertical shift of each function.

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- Sketch one cycle of each function on graph paper.
- Is the graph of is the same as the graph of ? Justify your answer

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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.

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- The musical note A above middle C on a piano makes a sound that can be modeled by the sine wave , where x represents time in seconds, and y represents the sound pressure. What is the period of this function?
- 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.

Homework

Use the functions below to answer questions 125– 128.

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- Find the amplitude of each function.
- Find the period of each function.
- Find the phase shift of each function.
- Find the vertical shift of each function.

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- Sketch one cycle of each function on graph paper.

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State the amplitude, period, phase shift, and vertical shift for each function. Draw the graph by hand and check with a graphing calculator.

130.

131.

132.

133.

134.

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

INSTRUCTION 1 :KHAN ACADEMYINSTRUCTION 2 SOPHIA

Class Work

Simplify the expression

138. 139.

141.

143.

144. 145.

Verify the Identity

147.

148. 149.

Homework

Simplify the expression

150. 151. 152.

153. 154.

155. 156.

157. 158.

Verify the Identity

160.

161. 162.

PERIODIC TRIG UNIT REVIEW

Multiple Choice

- How many degrees is
- 160
- 110
- 80
- 62
- Which angle is
- c.
- d.
- Which of the following angles is/are co-terminal with 170 (choose all correct answers)?

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- 340
- 190
- -190
- 530

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- Which is larger and by how much: an angle of 258, or an angle of radians?

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- 258 by
- 258 by radian
- radians by
- radians by

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- The central angle of a circle has a measure of radians and it intercepts an arc whose length is 5 meters. What is the approximate length in meters of the radius of the circle?

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- 19.6 m
- 2.0 m
- 1.3 m
- 12.6 m

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- is the radian measure of a central angle that intercepts an arc of length in a circle with a radius . If and r = 9, what is the value of s?

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- 18.8
- 4.3
- 0.23
- 56.5

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- 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?

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- 1,119,744 inches
- 36 inches
- 24 inches
- 576 inches

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- Given the terminal point of find

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- -1
- 1

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- Knowing and the terminal point is in the second quadrant find

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

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- What is the phase shift of

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- Name the amplitude and vertical shift of
- Amplitude: -0.5, Vertical Shift: -3
- Amplitude: 0.5, Vertical Shift: -3
- Amplitude: , Vertical Shift: 3
- Amplitude: , Vertical Shift: -3

- Which graph represents ?
- c.
- d.
- The difference between the maximum of and is

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- 1
- 2
- 3
- 8

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- Find the exact value of

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- On the interval , if , what is ?

- 0
- all of the above

- If the angle is placed in standard position, its terminal side lies in quadrant II and What is the value of (This problem is from the NJ Model Curriculum assessment for Algebra II Unit 3.)
- c. 0.75
- d. 0.8

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Extended Response

- Sketch the graph of
- 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.
- Which trig function would be the best fit for this model (assuming 9AM is t=0)?
- Write the equation that models this situation.
- Determine the amplitude, period, and midline.
- Predict the water level at midnight.

21. Cal C. was asked to solve the following equation over the interval . During his calculations he might have made an error. Identify the error and correct his work so that he gets the right answer.

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