Question

If $\left[\sin x\right]+\left[\sqrt{2}\cos x\right]=-3,x\in [0,2\pi ]$ $([.]$ denotes the greatest integer function $)$ then prove that $x$ belongs to $(\pi ,\frac{5\pi }{4})$,

Answer 1

We have $\left[\sin x\right]+\left[\sqrt{2}\cos x\right]=-3$
$\Rightarrow \left[\sin x\right]=-1$ and $\left[\sqrt{2}\cos x\right]=-2$
$\Rightarrow -1\le \sin x\le 1\Rightarrow x\le (\pi ,2\pi )$
and $-2\le \sqrt{2}\cos x<-1\Rightarrow \cos x<-\frac{1}{\sqrt{2}}\Rightarrow x\in (\frac{3\pi }{4},\frac{5\pi }{4})$
So $x\in (\pi ,\frac{5\pi }{4})$