Question

The period of $\sin \frac{\pi \left[x\right]}{12}+\cos \frac{\pi \left[x\right]}{4}+\tan \frac{\pi \left[x\right]}{3}$ where $\left[x\right]$ represents the greatest integer less than or equal to $x$ is

• A. $12$
• B. $4$
• C. $3$
• D. $24$

Answer 1

The correct option is D $24$

Since the period of $\sin \left(x\right)$ is $2\pi$, the period of $\sin \left(\frac{\pi \left[x\right]}{12}\right)$, where $\left[x\right]$ is a greatest integer function, is $\frac{2\pi }{\frac{\pi }{12}}=24$.

period of $tanx$ is $\pi$ and period of $cosx$ is $2\pi$
Similarly, the periods of $\cos \left(\frac{\pi \left[x\right]}{4}\right)$ and $\tan \left(\frac{\pi \left[x\right]}{3}\right)$ are $8$ and $3$ respectively.
Hence the period of the function $\sin \left(\frac{\pi \left[x\right]}{12}\right)+\cos \left(\frac{\pi \left[x\right]}{4}\right)+\tan \left(\frac{\pi \left[x\right]}{3}\right)$ is the LCM of the periods of the three functions added.
Hence the period of the given function is LCM $(24,8,3)=24$.