The period of f(x)=cos left(frac{x}{3}right)+ | vedclass

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Question

(D) Given,$f(x)=\cos \left(\frac{x}{3}\right)+\sin \left(\frac{x}{2}\right)$.We know that the period of $\cos(ax)$ and $\sin(ax)$ is $\frac{2\pi}{|a|}

Vedclass · Accepted Answer

$12$

Vedclass · Answer

(D) Given,$f(x)=\cos \left(\frac{x}{3}ight)+\sin \left(\frac{x}{2}ight)$.We know that the period of $\cos(ax)$ and $\sin(ax)$ is $\frac{2\pi}{|a|}$.For the term $\cos \left(\frac{x}{3}ight)$,the period $T_1 = \frac{2\pi}{1/3} = 6\pi$.For the term $\sin \left(\frac{x}{2}ight)$,the period $T_2 = \frac{2\pi}{1/2} = 4\pi$.The period of the sum of two periodic functions is the Least Common Multiple $(LCM)$ of their individual periods.Therefore,the period of $f(x) = 	ext{LCM}(6\pi, 4\pi)$.Since $6\pi = 2 	imes 3\pi$ and $4\pi = 2 	imes 2\pi$,the $	ext{LCM}(6\pi, 4\pi) = 12\pi$.Thus,the period of $f(x)$ is $12\pi$.

The period of $f(x)=\cos \left(\frac{x}{3}\right)+\sin \left(\frac{x}{2}\right)$ is (in $\pi$)

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