Graphs of Sine and Cosine Graphs of Sine and Cosine FunctionsFunctionsSection 4.5 Section 4.5

Fast Wave, Slow WaveFast Wave, Slow Wave

Who cares about waves??Who cares about waves??

Create the sine and Create the sine and cosine Graphscosine GraphsCreating the Graphs

Angle Value Sine Cosine0° (0 radians) 0 1

30° (π/6 radians) ½ √3/245° (π/4 radians) √2/2 √2/260° (π/3 radians) √3/2 1/290° (π/2 radians) 1 0

135° (3π/4 radians)

√2/2 -√2/2

180° (π radians) 0 -1270° (3π/2

radians)-1 0

360° (2π radians) 0 1450° (5π/2 radians) -1 0540° (2π radians) 0 -1630° (2π radians) -1 0720° (2π radians) 0 1

sine and cosine sine and cosine graphsgraphs

Sine graph plotted Sine graph plotted 

Cosine graph plotted Cosine graph plotted 

What about negative What about negative values?values?

Does the graph repeat?Does the graph repeat?

When?When?

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

Key PointsKey PointsWhat did you find???What did you find???

Key PointsKey Points

What are key points? What do they What are key points? What do they mean?mean?

Review – Graphing sine and Review – Graphing sine and cosinecosine

• Where do they graphs come from?!?Where do they graphs come from?!?

• Graphing Sine and Cosine – ReviewGraphing Sine and Cosine – Review

• graphing sine and cosine - radiansgraphing sine and cosine - radians

Listing the Key PointsListing the Key Points

Sinx:(0°,0) (90°, 1) (180°,0) (270°, -1) Sinx:(0°,0) (90°, 1) (180°,0) (270°, -1) (360°, 0)(360°, 0)

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

Transformations to the Transformations to the parent graphs – sine and parent graphs – sine and

cosinecosine• We will see how transformations We will see how transformations

affect the basic sine and cosine affect the basic sine and cosine parent graphsparent graphs

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

• y = acos(bx – c) + dy = acos(bx – c) + d

Vertical Vertical TranslationsTranslations• A vertical shift is the A vertical shift is the verticalvertical distance distance

between the midline of the graph and the between the midline of the graph and the x x ––axis.axis.

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

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

What’s What’s Happening??Happening??

Transformations of Sine and Cosine Transformations of Sine and Cosine 

Vertical Vertical TranslationsTranslations

• The shift is d units upward for d >0 The shift is d units upward for d >0

• The shift is d units downward for d< 0 The shift is d units downward for d< 0

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

• Does the period change when “d” changes?Does the period change when “d” changes?

Examples:Examples:

AmplitudeAmplitude•The amplitude of y = aThe amplitude of y = asinsinx and y x and y

= a= acoscosx represents the x represents the verticalvertical distance between the midline distance between the midline and the maximum or minimumand the maximum or minimum

•Amplitude = |a|Amplitude = |a|

•The constant factor “a” is a The constant factor “a” is a scaling factor - a vertical scaling factor - a vertical stretchstretch or or shrinkshrink of the basic sine and of the basic sine and cosine curvescosine curves

What’s What’s Happening??Happening??

Transformations of Sine and Cosine Transformations of Sine and Cosine 

Amplitude…What Does It Amplitude…What Does It Do??Do??

• If a ≥ 1, the basic sine curve is If a ≥ 1, the basic sine curve is stretched verticallystretched vertically

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

•The graph of y = aThe graph of y = asinsinx ranges between x ranges between -a and a instead of between -1 and 1-a and a instead of between -1 and 1

Amplitude…What Does It Amplitude…What Does It Do??Do??

• If 0 <a < 1, the basic sine curve is If 0 <a < 1, the basic sine curve is vertically shrunkvertically shrunk

• If -1< a < 0, the basic sine curve is If -1< a < 0, the basic sine curve is reflected across the x-axis and reflected across the x-axis and vertically shrunkvertically shrunk

•The graph of y = aThe graph of y = asinsinx ranges between x ranges between -a and a instead of between -1 and 1-a and a instead of between -1 and 1

Examples:Examples: