**6.4 Graphs of Sine and Cosine Functions**

The Sine Graph

This is the graph of

y = sin x

it is the parent graph for all sine functions.

The Cosine Graph

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Real Life Application

Electricity can flow in both an alternating current (AC) or a direct current (DC). Alternating currents are very interesting because they flow in the form of a sine wave. The AC is how electricity is delivered to house and businesses. An oscilloscope can measure oscillations of an electrical current and display it on a screen in the form of a sine graph

The parent graph has five key points: 0, π/2, π, 3π/4, and 2π.

0

This video gives the unit circle definition of the sine graph.

This video shows how to graph sine and cosine functions.

This is the graph of

y = cos x

it is the parent graph for all cosine functions.

The parent graph has five key points: 0, π/2, π, 3π/4, and 2π.

Multiple Choice Problem

Write the equation of the sine function given the characteristics.

Answers

Graphing Transformations

If

w

> 0, the amplitude and period of

y =

A

sin (

w

x) and y =

A

cos (

w

x)

are given by:

Using our knowledge of transformations

y =

3

sin (

2

x)

has an amplitude of

3

and a period of

π

.

Amplitude = I

A

I Period = T = 2π/

w

Amplitude: 3

Period: 2

A. y = 3 sin (2x)

B. y = ±4 sin (πx)

C. y = ±3 sin (πx)

D. y = 4 sin (2x)

Correct Answer

C. y = ±3 sin (πx)

Free Response Problem

Graph the function

Solution

y = 2 sin (.5x)

Solution

Amplitude:

3

Period:

2

y = I

A

I sin (

w

x)

y = I

3

I sin (

π

x)

y = ±

3

sin (

π

x)

Remember

Period = 2π/

w

2 = 2π/

w

2

w

= 2π

w

= 2π/2

w

= π

First we have our parent graph y = sin x.

Then we change the parent graphs amplitude to

2

because our function was y =

2

sin (.5x).

Finally we adjust the period. The period is 4π because our function was

y = 2 sin (

.5

X).

Remember Period = T = 2π/

w

T = 2π/

.5

T = 4π

Theorem

Oscilloscope

**More Practice**

Page 407-409

Questions 27,35,65,73

Chinese Spouting Bowl

The Chinese Spouting Bowl was created during the Han Dynasty in ancient China. By rubbing ones hands on the handles a skilled person can make waves jump out of the bowl at up to three feet high.

It works by using sine graphs. Rubbing the handle of the bowl creates vibrations and in turn the vibrations create nodes and anti nodes. A node is a place of no displacement and and anti node is the opposite.

Two sine waves are created because there are two handles. The sine waves collide and create spouts of water. The following video explains the phenomenon pretty well.