Periodic Functions Math Example 1

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Example 1

easy

Verify that

f(x) = \sin(x)

is periodic with period

2\pi

by checking the definition

f(x + p) = f(x)

.

Solution

1
Recall the identity: $\sin(x + 2\pi) = \sin x \cos 2\pi + \cos x \sin 2\pi$ .
2
Substitute known values $\cos 2\pi = 1$ and $\sin 2\pi = 0$ : $\sin(x + 2\pi) = \sin x \cdot 1 + \cos x \cdot 0 = \sin x$ .
3
Since $f(x + 2\pi) = f(x)$ for all $x$ , and $2\pi$ is the smallest such positive number, the period is $p = 2\pi$ .

Answer

f(x) = \sin(x)

has period

2\pi

A function is periodic if it repeats its values at regular intervals. The sine function satisfies

f(x+2\pi)=f(x)

for all real

x

, making

2\pi

its fundamental period.

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.

Learn more about Periodic Functions →

More Periodic Functions Examples

Example 2 medium

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

The function [formula] has period [formula]. What is [formula] given that [formula]?

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

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