Question

Total number of solutions of $\left[\sin x\right]+\cos x=0;$ where$[.]$ denotes the greatest integer function, for $xϵ[0,2\pi ]$, is:

• A. 1
• B. 2
• C. 4
• D. 0

Answer 1

The correct option is D 0

Given, $\left[\sin x\right]+\cos x=0\Rightarrow \cos x=\left[\sin x\right]$

If $xϵ[0,\pi ]-\left(\pi /2\right)$

$\Rightarrow \left[\sin x\right]=0\Rightarrow \cos x=0$

So there is no solution in this interval

if $xϵ(\pi ,2\pi )-(-pi/2)$

$\Rightarrow \left[\sin x\right]=-1\Rightarrow \cos x=-1$

So there is no solution in this interval as well

Clearly $x=\frac{\pi }{2}$ and $x=2\pi$ are not solution

Hence there is no solution for given equation.