4.5 Graphs of Sine and Cosine Functions

• Students will sketch the graphs of basic sine and cosine functions.

• Students will use amplitude and period to help sketch the graphs of sine and cosine functions.

• Students will sketch translations of graphs of sine and cosine functions.

• Students will use sine and cosine functions to model real-life data.

Evaluate the Sine Curve using the unit circle

Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–17

Section 4.5, Figure 4.42, Graph of Sine Curve, pg. 287

x

y (0, 1)

90°2

(–1, 0)

180°

(0, –1)

270°32

(1, 0)0° 0

60° 3

31 ,2 2

45°4

2 2,2 2

30° 6

3 1,2 2

330°116

3 1,2 2

315°74

2 2,2 2

300°53

31 ,2 2

23

120°

31 ,2 2

34

135°

2 2,2 2

56

150°

3 1,2 2

210°76

3 1,2 2

225°5

4

2 2,2 2

240°43

31 ,2 2

The Sine Curve

Evaluate the Cosine Curve using the unit circle

Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–18

Section 4.5, Figure 4.43, Graph of Cosine Curve, pg. 287

x

y (0, 1)

90°2

(–1, 0)

180°

(0, –1)

270°32

(1, 0)0° 0

60° 3

31 ,2 2

45°4

2 2,2 2

30° 6

3 1,2 2

330°116

3 1,2 2

315°74

2 2,2 2

300°53

31 ,2 2

23

120°

31 ,2 2

34

135°

2 2,2 2

56

150°

3 1,2 2

210°76

3 1,2 2

225°5

4

2 2,2 2

240°43

31 ,2 2

The Cosine Curve

Section 4.5, Figure 4.44, Key Points in the Sine and Cosine

Curves, pg. 288

Graph y = sin x and y = 2 sin x on your graphing calculator. Notice that the height of the hump has changed. In the equation y = a sin x is known as the amplitude of the function.

a

Graph y = cos x and y = cos 2x on your graphing calculator. Notice that the length of the curve has changed. In the equation y = cos bx, b affects the period of the function. Using sin and cos P

b

2

Find the period and amplitude

p. 294 #1 y x3 2sin

Find the period and amplitude

p. 294 #11 yx

1

4

2

3cos

Describe the relationship between the graphs of f and g. Consider amplitudes, periods, and shifts.

p. 294 #15f x x

g x x

( ) sin

( ) sin( )

Describe the relationship between the graphs of f and g. Consider amplitudes, periods, and shifts.

p. 294 #21f x x

g x x

( ) sin

( ) sin

2

5 2

Sketch the graphs of f and g in the same coordinate plane. (Include two full periods.)

p. 294 #27f x x

g x x

( ) sin

( ) sin

4

Reference Graphs y = sin x y = cos x