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Chapter Chapter 4.3 B4.3 BCircular FunctionsCircular Functions

Sine and Cosine FunctionsSine and Cosine Functions

Horizontal Phase Shift:0 : units to the right.

0 : units to the left.

Vertical Phase Shift:0 : units upward.

0: units downward.

h h

h h

k k

k k

Example 4.3.4Example 4.3.4

For the following item,a. find the domain and range.b. find the amplitude.c. find the period.d. identify the horizontal phase shift.e. identify the vertical phase shift.f. sketch 1 cycle of the graph.

12sin 1 2sin 0 12 212 0 12

, 1,3

amplitude: 22 2

412

xf x x

a b h k

Dom f

Rng f k a k a

a

pb

horizontal phase shift: nonevertical phase shift: 1 unit upward.

0 1h k

04

hp

0 42 3

x 0 2 3 4

y

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x 0 2 3 4

y 1 -1 1 3 1

2sin 12

00 2sin 1 2 0 1 1

2

2sin 1 2 1 1 1222 2sin 1 2 0 1 12

33 2sin 1 2 1 1 3

24

4 2sin 1 2 0 1 12

xf x

f

f

f

f

f

x 0 2 3 4

y 1 -1 1 3 1

0 2 3 4

123

123

1,3

Dom f

Rng f

Subdivision of a PeriodSubdivision of a Period

In general, we may consider the followingsubdivision of one period.

h h p2p

h4ph 3

4p

h

Fundamental CurveFundamental Curve

Fundamental Domain: 0,2

2 : extend the graph until it coversthe fundamental domain.

2 : start at 0 and the graph shouldcontain one complete periodeven if it exceeds 2 .

p

p

Example 4.3.5Example 4.3.5

Sketch the fundamental curve of 2sin 1.2x

f x

0 2 3 4

123

123

4 2p

fundamental curve

Example 4.3.6Example 4.3.6

For the following item,a. find the domain and range.b. find the amplitude.c. find the period.d. identify the horizontal phase shift.e. identify the vertical phase shift.f. sketch 1 cycle of the graph.g. sketch the fundamental curve.

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3cos 2 3cos22

3cos22

3 2 02

, 3,3

amplitude: 3

22

g x x x

g x x

a b h k

Dom g

Rng g k a k a

a

p

horizontal phase shift: units to the left2vertical phase shift: none

02

h k

2

4 4

h

pp

2

20

4

4

x -/2 -/4 0 /4 /2

y

x -/2 -/4 0 /4 /2

y 3 0 -3 0 3

3cos 2

3cos 2 3cos 0 3 1 32 2

3cos 2 3cos 3 0 04 4 2

0 3cos 2 0 3cos 3 1 3

33cos 2 3cos 3 0 04 4 2

3cos 22 2

g x x

g

g

g

g

g

3cos 2 3 1 3

x -/2 -/4 0 /4 /2

y 3 0 -3 0 3

2

4

0

4

2

1

2

3

3

2

1

2p

4

2 3

4 5

4 3

2 7

4

1

2

3

22

4

3

2

1

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4sin 2 2 4sin2 22

4 2 22

, 2,6

amplitude: 422

h x x x

a b h k

Dom h

Rng h k a k a

a

p

horizontal phase shift: units to the right

2vertical phase shift: 2 units upward

22

h k

2

4 4

h

pp

2 3

23

4 5

4

x /2 3/4 5/4 3/2

y

x /2 3/4 5/4 3/2

y 2 6 2 -2 2

4sin 2 2h x x

2 3

4 5

4 3

2

123456

1

2

4

2 3

4 5

4 3

2 7

4 2

123

456

12

2,6

Dom h

Rng h