The trigonometric ratios can also be considered as functions of a variable which is the measure of an angle. This angle measure can either be given in degrees or radians . Here, we will use radians.
The graph of a sine function y = sin ( x ) is looks like this:
Properties of the Sine Function, y = sin ( x )
Domain : ( − ∞ , ∞ )
Range : < − 1 , 1 > or − 1 ≤ y ≤ 1
y -intercept : ( 0 , 0 )
x -intercept : n π , where n is an integer.
Period: 2 π
Continuity: continuous on ( − ∞ , ∞ )
Symmetry: origin (odd function)
The maximum value of y = sin ( x ) occurs when x = π 2 + 2 n π , where n is an integer.
The minimum value of y = sin ( x ) occurs when x = 3 π 2 + 2 n π , where n is an integer.
Amplitude and Period of a Since Function
The amplitude of the graph of y = a sin ( b x ) is the amount by which it varies above and below the x -axis.
Amplitude = | a |
The period of a sine function is the length of the shortest interval on the x -axis over which the graph repeats.
Period = 2 π | b |
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Example:
Sketch the graphs of y = sin ( x ) and y = 2 sin ( x ) . Compare the graphs.
For the function y = 2 sin ( x ) , the graph has an amplitude 2 . Since b = 1 , the graph has a period of 2 π . Thus, it cycles once from 0 to 2 π with one maximum of 2 , and one minimum of − 2 .
Observe the graphs of y = sin ( x ) and y = 2 sin ( x ) .
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Each has the same x -intercepts, but y = 2 sin ( x ) has an amplitude that is twice the amplitude of y = sin ( x ) .