http://www.math.utah.edu/~palais/sine.html
http://www.ies.co.jp/math/java/trig/index.html
http://www.analyzemath.com/function/periodic.html
http://math.usask.ca/maclean/SinCosSlider/SinCosSlider.html
http://www.analyzemath.com/unitcircle/unitcircle.html
http://biorhythms.perbang.dk/?aid=29&name=Laura&d=18&m=2&y=1979&t=23&min=54
&z=­5&phy=1&emo=1&inte=1&custom_lang=en&js=&ctrl=&custom=1&js=
http://www.sfu.ca/~jtmulhol/calculus­applets/GeoGebra­Worksheets/trigonometric­graphs.html
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Chapter 5: Trigonometric Functions and their Graphs
Section 5.1: Graphing Sine and Cosine Functions
Periodic Function: A function that repeats itself over regular intervals (cycles) of its domain. Sine and Cosine functions are periodic functions. The values of these functions repeat over a specified period.
Period
The horizontal length of one cycle on a periodic graph. (shortest part that repeats, measured along the horizontal axis)
http://www.analyzemath.com/fu
Sinusoidal curve
A curve that oscillates repeatedly up and down from a centre line. (looks like a wave) 2
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Periodic Functions
The simplest example of periodic motion is the motion around a circle.
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Point on a unit circle P(x, y)
P (x, y)
(Cos θ, Sin θ)
Any point on unit circle given as (x, y)
•
x corresponds to cosine of angle
• y corresponds to sine of angle
As P rotates around the circle the values of
cos θ and sin θ change periodically
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The arc length of cirlce can be used instead
of the angle at the centre as
the variable in the cos and sin functions.
x in the functions represents the length of the arc from A(1, 0) to an point on the circle.
• The cos x is the horizontal coordinate of P
• The sin x is the vertical coordinate of P
P (Cos x, Sin x)
A (1, 0)
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An arc length measured along the UNIT CIRCLE equals the measure of the central angle in radians. 12
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0.5
­0.5
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­0.5
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Read Page 224
http://www.analyzemath.com/unitcircle/unitcircle.html
http://www.math.utah.edu/~palais/sine.html
http://www.math.utah.edu/~palais/cossin.html
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Read Page 224
http://www.analyzemath.com/unitcircle/unitcircle.html
http://www.math.utah.edu/~palais/sine.html
http://www.math.utah.edu/~palais/cossin.html
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http://www.math.utah.edu/~palais/cossin.html
http://math.usask.ca/maclean/SinCosSlider/SinCosSlider.html
(x, y)
180
90
radius=1
270
360
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0.5
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EX: What is the amplitude of the following cosine function?
a) Y max= 6 Y min= ­4
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HORIZONTAL STRETCH "b'
If b > 1 the graph is compressed horizontally and the period is decreased
If 0< b <1 the graph is expanded horizontally and the period of the graph is increased
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To find the "b" value for making a equation look at the graph and find the period of the graph.
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AMPLITUDE
Amplitude is always positive because it represents distance from the max. or min. point of the graph to the midline (halfway point)
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http://www.analyzemath.com/function/periodic.html
http://www.math.utah.edu/~palais/sine.html
http://www.ies.co.jp/math/java/trig/index.html
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WRITE AN EQUATION OF A GRAPH
1. Find the MIDLINE of function (draw a dotted line)
­ Use the midline to find amplitude or vertical stretch
­ Use to find vertical displacement (the midline y value is the vertical displacement)
2. Find the period of the graph by looking at the graph and use to calculate "b"
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HOMEWORK
PAGE 233­236
#6, 7a,b, 8 a,b,c, 9, 14
10a,b, 4abc,5abc, 11bd
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