The Periods of the Sine and Cosine Functions

One thing we can observe from the graphs of the sine function in the beginning activity is that the graph seems to have a “wave” form and that this “wave” repeats as we move along the horizontal axis. We see that the portion of the graph between 0 and (2pi) seems identical to the portion of the graph between (2pi) and (4pi) and to the portion of the graph between (-2pi) and 0. The graph of the sine function is exhibiting what is known as a periodic property. Figure 2.1 shows the graph of (y = sin(t)) for three cycles.

Figure (PageIndex{1}): Graph of (y = sin(t)) with (-2pi leq t leq 4pi)

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We say that the sine function is a periodic function. Such functions are often used to model repetitious phenomena such as a pendulum swinging back and forth, a weight attached to a spring, and a vibrating guitar string.

The reason that the graph of (y = sin(t)) repeats is that the value of (sin(t)) is the y-coordinate of a point as it moves around the unit circle. Since the circumference of the unit circle is (2pi) units, an arc of length ((t + 2pi)) will have the same terminal point as an arc of length t. Since (sin(t)) is the y-coordinate of this point, we see that (sin(t + 2pi) = sin(t)). This means that the period of the sine function is (2pi). Following is a more formal definition of a periodic function.

Notice that if (f) is a periodic function with period (p), then if we add 2(p) to (t), we get [f(t + 2p) = f((t+p)+p) = f(t + p) =f(t).]

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We can continue to repeat this process and see that for any integer (k), [f(t + kp) =f(t).]

So far, we have been discussing only the sine function, but we get similar behavior with the cosine function. Recall that the wrapping function wraps the number line around the unit circle in a way that repeats in segments of length (2pi). This is periodic behavior and it leads to periodic behavior of both the sine and cosine functions since the value of the sine function is the (y)-coordinate of a point on the unit circle and the value of the cosine function is the (x)-coordinate of the same point on the unit circle, the sine and cosine functions repeat every time we make one wrap around the unit circle. That is, [cos(t + 2pi) = cos(t) space and space sin(t + 2pi) = sin(t).] It is important to recognize that (2pi) is the smallest number that makes this happen. Therefore, the cosine and sine functions are periodic with period (2pi).

Figure (PageIndex{2}): Graph of (y = cos(t)) with (-2pi leq t leq 4pi)

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