Periodic Functions Math Example 3

Follow the full solution, then compare it with the other examples linked below.

easy

The function

h(x) = \tan(x)

has period

\pi

. What is

\tan\!\left(\frac{5\pi}{4}\right)

given that

\tan\!\left(\frac{\pi}{4}\right) = 1

1
Use periodicity: $\tan\!\left(\frac{5\pi}{4}\right) = \tan\!\left(\frac{\pi}{4} + \pi\right)$ .
2
Since the period is $\pi$ , $\tan\!\left(\frac{\pi}{4} + \pi\right) = \tan\!\left(\frac{\pi}{4}\right) = 1$ .

Answer

\tan\!\left(\dfrac{5\pi}{4}\right) = 1

Periodicity lets us reduce any angle to an equivalent one in the fundamental period. Because

\tan

repeats every

\pi

radians, adding

\pi

to the argument leaves the value unchanged.

About Periodic Functions

A function that repeats its values at regular intervals: $f(x + T) = f(x)$ for all $x$ , where $T$ is the smallest positive period.

Verify that [formula] is periodic with period [formula] by checking the definition [formula].

Find the period of [formula] and sketch one complete cycle.

A function satisfies [formula] for all [formula], and is defined on [formula] by [formula]. Find [fo