# Graphs of Sine and Cosine Functions

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16-Mar-2016Category

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Graphs of Sine and Cosine FunctionsSection 4.5

Fast Wave, Slow Wave

Who cares about waves??

Create the sine and cosine Graphs

Creating the Graphs

Angle ValueSineCosine0 (0 radians)0130 (/6 radians)3/245 (/4 radians)2/22/260 (/3 radians)3/21/290 (/2 radians)10 135 (3/4 radians)2/2-2/2180 ( radians)0-1270 (3/2 radians)-10360 (2 radians)01450 (5/2 radians)-10540 (2 radians)0-1630 (2 radians)-10720 (2 radians)01

sine and cosine graphs

Sine graph plotted Cosine graph plotted

What about negative values?

Does the graph repeat?

When?

One complete cycle of the cosine and sine graphs is called the ___period___

Key PointsWhat did you find???

Key Points

What are key points? What do they mean?

Review Graphing sine and cosineWhere do they graphs come from?!?

Graphing Sine and Cosine Review

graphing sine and cosine - radians

Listing the Key PointsSinx:(0,0) (90, 1) (180,0) (270, -1) (360, 0)

Cosx:(0,1) (90,0) (180, -1) (270, 0) (360, -1)

Transformations to the parent graphs sine and cosineWe will see how transformations affect the basic sine and cosine parent graphs

General Equations:y = asin(bx c) + d

y = acos(bx c) + d

Vertical TranslationsA vertical shift is the vertical distance between the midline of the graph and the x axis.

For y = sinx +d and y = cosx +d, the constant d causes a vertical shift in the graph

Which value (x or y) is influenced by the change in the d value?

Whats Happening?? Transformations of Sine and Cosine

Vertical Translations

The shift is d units upward for d >0

The shift is d units downward for d< 0

The graph oscillates about line y = d instead of x-axis

Does the period change when d changes?

Examples:

AmplitudeThe amplitude of y = asinx and y = acosx represents the vertical distance between the midline and the maximum or minimum

Amplitude = |a|

The constant factor a is a scaling factor - a vertical stretch or shrink of the basic sine and cosine curves

Whats Happening?? Transformations of Sine and Cosine

AmplitudeWhat Does It Do??If a 1, the basic sine curve is stretched vertically

If a -1, the basic sine curve is reflected across the x-axis and vertically stretched

The graph of y = asinx ranges between -a and a instead of between -1 and 1

- AmplitudeWhat Does It Do??If 0
Examples:

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