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CHAPTER 4 – LESSON CHAPTER 4 – LESSON 11
How do you graph sine and cosine by unwrapping the unit circle?
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Warm-Up/ActivatorWarm-Up/ActivatorFill in the table (separate sheet)
with the radian measure of the angles and then both the exact and approximate values for sine, cosine, and tangent of these angles.
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Angle Chart for Unit Circle
0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360
0 30 45 60 90 0 30 45 60 9060 45 30 60 45 30 0
2
10 2
2
2
31 2
10 2
22
3-1 02
3
2
32
2
2
22
1
2
1
1 02
3
2
2
2
1
2
1
2
2
2
312
12
22
3-1 02
32
22
1
0 3
11 3 -- 0 3
11 3 -- 3
3
1-1 03
3
1-1
33
110 0-- -- -- 3
3
113 1 3
1 3 1 3
1
1 -- -1 -- 123
22 2
3
22
3
222
3
222
-- 13
222 -- 2
3
22 -1 -- 2
3
22
3
222
0 .5 .707 .866 1 .866 .707 .5 0 -.5 -.707 -.866 -1 -.866 -.707 -.5 0
1 .866 .707 .5 0 -.5 -.707 -.866 -1 -.866 -.707 -.5 0 .5 .707 .866 1
0 .577 1 1.7 -- -1.7 -1 -.577 0 .577 1 1.7 -- -1.7 -1 -.577 0
0 -.577 -1 -1.7 -- 1.7 1 .577 0 -.577 -1 -1.7 ---- 1.7 1 .577
1 1.15 1.414 2 -- -2 -1.414 -1.15 -1 -1.15 -1.414 -2 -- 2 1.414 1.15 1
-- 2 1.414 1.15 1 1.15 1.414 2 -- -2 -1.414 -1.15 -1 -1.15 -1.414 -2 --
30-60-90 45-45-90
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Graphs of FunctionsGraphs of FunctionsSine
30 60 90 120 150 180 210 240 270 300 330 360
-1.5
-1
-0.5
0.5
1
1.5
x
y
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Graphs of FunctionsGraphs of FunctionsCosine
30 60 90 120 150 180 210 240 270 300 330 360
-1.5
-1
-0.5
0.5
1
1.5
x
y
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Graphs of FunctionsGraphs of FunctionsTangent
30 60 90 120 150 180 210 240 270 300 330 360
-1.5
-1
-0.5
0.5
1
1.5
x
y
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Graphs of FunctionsGraphs of FunctionsSine
Cosine
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Graphs of FunctionsGraphs of FunctionsTan
Cotangent
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Graphs of FunctionsGraphs of FunctionsSecant
Cosecant
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Chapter 4 - Lesson 2Chapter 4 - Lesson 2Transforming Trig Transforming Trig FunctionsFunctionsEssential Question:How can we use the amplitude, period, phase shift and vertical shift to transform the sine and cosine curves?Key Question:How do the values of A, B, H, and K impact the shape of the trigonometric functions?
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Warm-Up/ActivatorWarm-Up/ActivatorComplete the Exploring Sine
Graphs Activity and Report findings to the class.
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Alternate ActivatorAlternate ActivatorGraph each equation without a
calculator
Y = 2(x -3)2 + 1 y = - (x + 2)2 - 3
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Transformations:Transformations:Vertical Shift: the vertical
movement of the graph (“new” x-axis)Phase Shift: the horizontal
movement of the graph (“new” y-axis)Period: the number of degrees or
radians required to draw one complete cycle of the curve
Amplitude: the distance the curve is from the “new” x-axis
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Transformation EquationTransformation Equation
KHBAy )sin(
Amplitude and Inversion
Period
Combine to give Horizontal Movement
Vertical Movement
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TransformationsTransformationsVertical shift– KPhase shift– the opp of H/BPeriod– sin and cos 360/B
tan 180/B Amplitude-- |A|
the sign indicates if it is inverted
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Example 1Example 1y = 2 cos (3x)
amp = period =phase = vertical =
-π π 2π 3π 4π 5π 6π 7π
-4
-3
-2
-1
1
2
3
4
x
y
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Example 2Example 2y = cos (1/3x)
amp = period =phase = vertical =
-π π 2π 3π 4π 5π 6π 7π
-4
-3
-2
-1
1
2
3
4
x
y
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Example 3Example 3y = cos(4x) + 2
amp = period =phase = vertical =
-π π 2π 3π 4π 5π 6π 7π
-4
-3
-2
-1
1
2
3
4
x
y
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Example 4Example 4y = cos(x+Π) + 1
amp = period =phase = vertical =
-π π 2π 3π 4π 5π 6π 7π
-4
-3
-2
-1
1
2
3
4
x
y
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Example 5Example 5y = 3 sin(2x – Π) + 1
amp = period =phase = vertical =
-π π 2π 3π 4π 5π 6π 7π
-4
-3
-2
-1
1
2
3
4
x
y
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Example 6Example 6 y = -sin(4x) – 2
amp = period =phase = vertical =
-π π 2π 3π 4π 5π 6π 7π
-4
-3
-2
-1
1
2
3
4
x
y
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Example 7Example 7y = ½ cos(2x) + 2
amp = period =phase = vertical =
-π π 2π 3π 4π 5π 6π 7π
-4
-3
-2
-1
1
2
3
4
x
y
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Example 8Example 8 y =2 cos(1/2x+Π) – 1
amp = period =phase = vertical =
-π π 2π 3π 4π 5π 6π 7π
-4
-3
-2
-1
1
2
3
4
x
y
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Chapter 4 - Lesson 3Chapter 4 - Lesson 3Sinusoidal RegressionsSinusoidal RegressionsEssential Question:How can sinusoidal regressions be used to model periodic data?Key Question:How do you use the calculator to find sinusoidal regressions?
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