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 ) .