Graphs of Sine and Cosine

    This topic presents the graphs of the sine and cosine functions. These graphs can be obtained by plotting points or using a knowledge of the values of the trigonometric functions, reference angles, and symmetry. The basic transformations of the sine and cosine are detailed including the amplitude (height), the period (portion that repeats), and the phase shift (horizontal shift).

Proposition (Sine Graph) The sine function has the following graph




Notice that the sine function has zeros at (written as where is any integer) and obtains its maximum and minimum value of and at

Proposition (Cosine Graph) The cosine function has the following graph



Notice that the cosine function has zeros at and obtains its maximum and minimum value of and at

Proposition (Graphs of Sine and Cosine) Amplitude. If or then the graph has amplitude

Example (Graphs of Sine and Cosine) (a) Find the amplitudes of and and sketch the graphs.

    Solution. The amplitudes of both are and here are the graphs,






Example (Graphs of Sine and Cosine) Find the amplitudes of and and sketch the graphs.

    Solution. The amplitudes of both are and here are the graphs,

Proposition (Graphs of Sine and Cosine) Period. If or then the graph has period

Example (Period of Sine and Cosine) Find the period of and sketch the graph.

    Solution. The period is and here is the graph:




Example (Graphs of Sine and Cosine) Find the period of , the amplitude, and sketch a graph.

    Solution. The period is and the amplitude is Here is the graph:

Proposition (Graphs of Sine and Cosine) Phase Shift. If or then the graph has phase shift of

Example (Graphs of Sine and Cosine) Find the the phase shift, amplitude and period of and sketch the graph.

    Solution. The amplitude is 5, the period is the phase shift is Here is the graph:




Example (Graphs of Sine and Cosine) Find the the phase shift, amplitude and period of and sketch the graph.

    Solution. The amplitude is the period is the phase shift is Here is the graph:
    

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Cite this as:
Graphs Of Sine And Cosine
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/graphs-of-sine-and-cosine.html