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CHAPTER 4 – LESSON CHAPTER 4 – LESSON 11
How do you graph sine and cosine by unwrapping the unit circle?
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.
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
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
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
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
Graphs of FunctionsGraphs of FunctionsSine
Cosine
Graphs of FunctionsGraphs of FunctionsTan
Cotangent
Graphs of FunctionsGraphs of FunctionsSecant
Cosecant
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?
Warm-Up/ActivatorWarm-Up/ActivatorComplete the Exploring Sine
Graphs Activity and Report findings to the class.
Alternate ActivatorAlternate ActivatorGraph each equation without a
calculator
Y = 2(x -3)2 + 1 y = - (x + 2)2 - 3
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
Transformation EquationTransformation Equation
KHBAy )sin(
Amplitude and Inversion
Period
Combine to give Horizontal Movement
Vertical Movement
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
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
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
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
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
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
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
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
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
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?
Your TurnYour Turn
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