Question

The time period of ${\sin ^3}x + {\cos ^3}x$ is
(A) $\dfrac{\pi }{3}$
(B) $\pi$
(C) $2\pi$
(D) $\dfrac{{2\pi }}{3}$

Answer 1

First, we will determine the time period ${\sin ^3}x$ and the time period of ${\cos ^3}x$. Then, by using this we will determine the time period of the required function ${\sin ^3}x + {\cos ^3}x$ by taking L.C.M of the time period of ${\sin ^3}x$ and ${\cos ^3}x$ and dividing it by H.C.F. of 1 and 1.

Complete step-by-step answer:
Given the function is ${\sin ^3}x + {\cos ^3}x$.
The time period of ${\sin ^a}\theta$ (when a is odd) is $2\pi$.
Therefore, the time period of ${\sin ^3}x$ is $2\pi$.
The time period of ${\cos ^a}\theta$ (when a is odd) is $2\pi$.
Therefore, the time period of ${\cos ^3}x$ is $2\pi$.
The time period of ${\sin ^3}x + {\cos ^3}x$ is given by taking L.C.M of the time period of both the given functions ${\sin ^3}x$ and ${\cos ^3}x$ is given by
$\Rightarrow \dfrac{{L.C.M.(2\pi ,2\pi )}}{{H.C.F.(1,1)}}$
Since, the L.C.M. of $2\pi$ and $2\pi$ is $2\pi$. And the H.C.F. of 1 and 1 is 1. Therefore,
$\Rightarrow \dfrac{{L.C.M.(2\pi ,2\pi )}}{{H.C.F.(1,1)}} = \dfrac{{2\pi }}{1} = 2\pi$
Hence, the time period of ${\sin ^3}x + {\cos ^3}x$ is $2\pi$.

Note: The time period of ${\sin ^a}\theta$ ( when a is even) is $\pi$ and the time period of ${\cos ^a}\theta$ ( when a is even) is $\pi$. The time period of ${\sin ^a}\theta$ ( when a is odd) is $2\pi$ and the time period of ${\cos ^a}\theta$ ( when a is odd) is $2\pi$.