6.3

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6.3 Graphing Sine and Cosine Functions
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Periodic Function and Period

• A function is periodic if, for some real number
  a, f(x a) f(x) for each x in the domain of
  f.
• The least positive value of a for which
  f(x) f(x a) is the period of the function.

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Which function is periodic?
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Determine the period of the function
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Graph of Sine

• Look at the value for sin(x) for the domain value
  between -2p and 2p in multiples of p/4

x sin(x)
-2p 0
-7p/4 v2/2
-3p/2 1
-5p/4 v2/2
-p 0
-3p/4 -v2/2
-p/2 -1
-p/4 -v2/2
x sin(x)
0 0
p/4 v2/2
p/2 1
3p/4 v2/2
p 0
5p/4 -v2/2
3p/2 -1
7p/4 -v2/2
2p 0
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Graph of Sine
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Properties of the Graph of Sine

• What is the domain?
• What is the range?
• What is the period?
• Where are all the x-intercepts?
• Where is the y-intercept?
• Where are the maximums?
• Where are the minimums?

All real numbers
Set of real numbers between -1 and 1
2p
at pn, where n is any integer
at (0, 0)
At y 1 and occur when
x p/2 2pn
At y -1 and occur when
x 3p/2 2pn
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Find each value by referring to the graph of sine

1. sin(5p/2)
2. sin(3p)

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Find the values of ? for which sin ? -1 is
true.
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Graph y sinx for the interval of 2p to 4p
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Graph of Cosine

• Look at the value for cos(x) for the domain value
  between -2p and 2p in multiples of p/4

x cos(x)
-2p 1
-7p/4 v2/2
-3p/2 0
-5p/4 -v2/2
-p -1
-3p/4 -v2/2
-p/2 0
-p/4 v2/2
x cos(x)
0 1
p/4 v2/2
p/2 0
3p/4 -v2/2
p -1
5p/4 -v2/2
3p/2 0
7p/4 v2/2
2p 1
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Graph of Cosine
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Properties of the Graph of Cosine

• What is the domain?
• What is the range?
• What is the period?
• Where are all the x-intercepts?
• Where is the y-intercept?
• Where are the maximums?
• Where are the minimums?

All real numbers
Set of real numbers between -1 and 1
2p
at p/2 pn, where n is any
integer
at (0, 1)
At y 1 and occur when x
pn, n is an even integer
At y -1 and occur when x pn, n
is an odd integer
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Graphing in your calculator

• Make sure you are in radians!
• Type equation into the y s menu
• Go to Zoom Trig before you graph
• Then your graph will appear accurately

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Homework change

• Do NOT do 51, change it to 52