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How do you graph sine and cosine by unwrapping
the unit circle?
CHAPTER 4 – LESSON 1
Warm-Up/Activator

Fill 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.
30-60-90
Angle Chart for Unit Circle
0
30
45
60
90
120
135
150
180
210
225
0
30
45
60
90
60
45
30
0
30
45
0
1
2
2
2
3
2
1
3
2
2
2
1
2
0
1
2
 2
2
0
.5
.707
.866
1
.866
.707
.5
0
-.5
-.707 -.866
1
3
2
2
2
1
2
0
1
2
 2
2
1
.866
.707
.5
0
-.5
-.707 -.866
1
3
0
0
.577
--1
1
1
1
3
1.7
2
3
1.7
1
1
2
--
2
1
3
.577
2
1.15 1.414
--
3
2
1.414
2
2
2
3
1.15
--0
 3
-1.7
1
3
-1
-1
1
0
-.577 -1
--
2
--
-2
1
1
2
3
 2
 3
2
-1
 3
2
0
1
3
-.577
0
.577
 3
--
3
-1.7
--
1.7
2
3
-1
2
3
2
2
2
-1
---
 2
2
-.866 -.707
1
3
-1.414 -1.15
1.15 1.414
-1
1
1
1
1
240
270
300
315
330
360
60
90
60
45
30
0
 3
2
-1
 3
2
 2
2
-2
-.866 -.707
0
3
2
1
1
2
-.5
0
.5
.707
.866
 3
-1
1
3
--
3
1.7
--
-1.7
2
2
-1
0
-.5
0
1
0
-.577
0
 3
--
-1.7
--
1
3
0
.577
0
2
--
2
2
2
3
1
--
2
1.414
1.15
1
 2
2
--
 2
 2
-1
1
2
1
2
-1.15 -1.414 -2
2
45-45-90
2
3
-1.414 -1.15
-1
-1
1
3
1
-.577 -1
2
3
-1.15 -1.414 -2
--
Graphs of Functions

Sine
y
1.5
1
0.5
x
30
-0.5
-1
-1.5
60
90
120
150
180
210
240
270
300
330
360
Graphs of Functions

Cosine
y
1.5
1
0.5
x
30
-0.5
-1
-1.5
60
90
120
150
180
210
240
270
300
330
360
Graphs of Functions

Tangent
y
1.5
1
0.5
x
30
-0.5
-1
-1.5
60
90
120
150
180
210
240
270
300
330
360
Graphs of Functions

Sine

Cosine
Graphs of Functions

Tan

Cotangent
Graphs of Functions

Secant

Cosecant
Chapter 4 - Lesson 2
Transforming Trig Functions
Essential 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/Activator

Complete the Exploring Sine Graphs
Activity and Report findings to the class.
Alternate Activator

Graph each equation without a calculator

Y = 2(x -3)2 + 1
y = - (x + 2)2 - 3
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 Equation
Period
Vertical
Movement
y  A sin( B  H )  K
Amplitude and
Inversion
Combine to give
Horizontal
Movement
Transformations

Example 1

y
y = 2 cos (3x)
4
3
2
1
x
-π
π
2π
3π
-1
-2
-3
-4
amp =
 phase =

period =
vertical =
4π
5π
6π
7π
Example 2

y
4
y = cos (1/3x)
3
2
1
x
-π
π
2π
3π
-1
-2
-3
-4
amp =
 phase =

period =
vertical =
4π
5π
6π
7π
Example 3
y
4

y = cos(4x) + 2
3
2
1
x
-π
π
2π
3π
-1
-2
-3
-4
amp =
 phase =

period =
vertical =
4π
5π
6π
7π
Example 4
y
4

y = -sin(4x) – 2
3
2
1
x
-π
π
2π
3π
-1
-2
-3
-4
amp =
 phase =

period =
vertical =
4π
5π
6π
7π
Example 5
y
4

y = cos(x+Π) + 1
3
2
1
x
-π
π
2π
3π
-1
-2
-3
-4
amp =
 phase =

period =
vertical =
4π
5π
6π
7π
Example 6
y

4
y = 3 sin(2x – Π) + 1
3
2
1
x
-π
π
2π
-1
-2
-3
-4
amp =
 phase =

period =
vertical =
3π
4π
5π
6π
7π
Example 7
y
4

y = ½ cos(2x) + 2
3
2
1
x
-π
π
2π
3π
-1
-2
-3
-4
amp =
 phase =

period =
vertical =
4π
5π
6π
7π
Example 8
y
4

y =2 cos(1/2x+Π) – 1
3
2
1
x
-π
π
2π
3π
-1
-2
-3
-4
amp =
 phase =

period =
vertical =
4π
5π
6π
7π
Example 9 : Degrees
y
5
y =1.5 cos(1/2x+90) +2
4
3
2
1
x
-540
-450
-360
-270
-180
-90
90
-1
-2
-3
-4
-5
amp =
 phase =

period =
vertical =
180
270
360
450
540
Chapter 4 - Lesson 3
Sinusoidal Regressions
Essential Question:
How can sinusoidal regressions be used to model
periodic data?
Key Question:
How do you use the calculator to find sinusoidal
regressions?
Your Turn