6.3 – GRAPHING SINE AND
COSINE FUNCTIONS
Periodic Function and Period
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A function is periodic if, for some real number α,
f(x + α) = f(x) for each x in the domain of f.
The least positive value of α for which
f(x) = f(x + α) is the period of the function.
Which function is periodic?
Determine the period of the function
Graph of Sine
Look at the value for sin(x) for the domain value
between -2π and 2π in multiples of π/4
x
sin(x)
x
sin(x)
-2π
0
0
0
-7π/4
√2/2
π/4
√2/2
-3π/2
1
π/2
1
-5π/4
√2/2
3π/4
√2/2
-π
0
π
0
-3π/4
-√2/2
5π/4
-√2/2
-π/2
-1
3π/2
-1
-π/4
-√2/2
7π/4
-√2/2
2π
0
Graph of Sine
Properties of the Graph of Sine
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What is the domain? All real numbers
What is the range? Set of real numbers between -1 and 1
What is the period? 2π
Where are all the x-intercepts? at πn, where n is any integer
Where is the y-intercept? at (0, 0)
Where are the maximums? At y = 1 and occur when
x = π/2 + 2πn
Where are the minimums?
At y = -1 and occur when
x = 3π/2 + 2πn
Find each value by referring to the
graph of sine
1.
sin(5π/2)
2.
sin(3π)
Find the values of θ for which
sin θ = -1 is true.
Graph y = sinx for the interval of 2π to 4π
Graph of Cosine
Look at the value for cos(x) for the domain value
between -2π and 2π in multiples of π/4
x
cos(x)
x
cos(x)
-2π
1
0
1
-7π/4
√2/2
π/4
√2/2
-3π/2
0
π/2
0
-5π/4
-√2/2
3π/4
-√2/2
-π
-1
π
-1
-3π/4
-√2/2
5π/4
-√2/2
-π/2
0
3π/2
0
-π/4
√2/2
7π/4
√2/2
2π
1
Graph of Cosine
Properties of the Graph of Cosine
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What is the domain? All real numbers
What is the range? Set of real numbers between -1 and 1
What is the period? 2π
Where are all the x-intercepts? at π/2 + πn, where n is any
integer
at
(0,
1)
Where is the y-intercept?
Where are the maximums? At y = 1 and occur when
x = πn, n is an even integer
Where are the minimums?
At y = -1 and occur when
x = πn, n is an odd integer
Graphing in your calculator
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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
Homework change
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Do NOT do #51, change it to #52