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Graphing Cosine Function

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 cosine function y = cos ( x ) is looks like this:

Properties of the Cosine Function, y = cos ( x ) .

Domain : ( , )

Range : [ 1 , 1 ] or 1 y 1

y -intercept : ( 0 , 1 )

x -intercept : ( n π 2 , 0 ) , where n is an integer.

Period: 2 π

Continuity: continuous on ( , )

Symmetry: y -axis (even function)

The maximum value of y = cos ( x ) occurs when x = 2 n π , where n is an integer.

The minimum value of y = cos ( x ) occurs when x = π + 2 n π , where n is an integer.

Amplitude and Period a Cosine Function

The amplitude of the graph of y = a cos ( b x ) is the amount by which it varies above and below the x -axis.

Amplitude = | a |

The period of a cosine function is the length of the shortest interval on the x -axis over which the graph repeats.

Period = 2 π | b |

Example:

Sketch the graphs of y = cos ( x ) and y = 2 cos ( x ) . Compare the graphs.

For the function y = 2 cos ( 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 = cos ( x ) and y = 2 cos ( x ) . Each has the same x -intercepts, but y = 2 cos ( x ) has an amplitude that is twice the amplitude of y = cos ( x ) .

Also see Trigonometric Functions .