![]() You know from graphing quadratic functions of the form that as you changed the value of a you changed the “width” of the graph. For example, the graph of on the interval is one cycle. Any part of the graph that shows this pattern over one period is called a cycle. The period is the length of the smallest interval that contains exactly one copy of the repeating pattern. If a function has a repeating pattern like sine or cosine, it is called a periodic function. ![]() Each one contains exactly one complete copy of the “hill and valley” pattern. The graph below shows four repetitions of a pattern of length. This pattern continues in both directions forever. That is, the graph of (or ) on the interval looks like the graph on the interval or or. You know that the graphs of the sine and cosine functions have a pattern of hills and valleys that repeat. This change does not affect the graphs they remain the same. So from this point forward, we’ll refer to these same functions as and. Often the sine and cosine functions are used in applications that have nothing to do with triangles or angles, and the letter x is used instead of for the input (as well as to label the horizontal axis). We used the variable previously to show an angle in standard position, and we also referred to the sine and cosine functions as and.
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