Question

Number of solution(s) of the equation $\sin x=\left[x\right]$ [where [$.$] denotes greatest integer function] is

• A. $1$
• B. $2$
• C. $0$
• D. Infinite

Answer 1

Answer: (B)

$2$

We know that L. H. S lies between $-1$ and $1$ and R. H. S is always an integer.

So the only possible solutions could be $-1$, $0$ or $1$

We see that for, $x=0$ and $x=\frac{\pi }{2}$,
$\sin 0=0$ and $\sin \frac{\pi }{2}=1$ and $\sin (-\frac{\pi }{2})=-1$
Also,
$\left[0\right]=0$ and $\left[\frac{\pi }{2}\right]=1$ and $[-\frac{\pi }{2}]=-2$
Furthermore, for any other value of $x$ that $\sin x$ takes integral values, we find that $-1>\left[x\right]>1$

Thus $0$ and $\frac{\pi }{2}$ satisfy the equation $\sin x=\left[x\right]$
Hence, option 'B' is correct.