Question

In Exercises 59 and \(60,\) show that the function is periodic and find its period. $$y=\sin ^{3} x$$

Short answer

The function \(y=\sin ^3 x\) is periodic and its period is \(2\pi\).

Step-by-step solution

Validate Periodicity

Let's validate the periodicity of the function. Here, the function is \(f(x) = \sin^3 x\). Now, the function will be termed periodic if \(f(x + P) = f(x)\), where P is the period. By substituting \(x + P\) in place of x in the function, we get \(f(x+P) = \sin^3(x+P)\).

Show that \(\sin^3(x+P) = \sin^3 x\)

Observe that sine function, \(\sin x\), is periodic with period \(2\pi\). That is \(\sin(x + 2\pi) = \sin x\). Hence \(\sin^3(x+2\pi) = \sin^3 x\). Therefore, the function \(y = \sin^3 x\) is periodic and its period P is \(2\pi\).