Question

The number of solutions of $[\sin x+\cos x]=3+[-\sin x]+[-\cos x]$(where $\left[\right]$ denotes the greatest interger function),$xϵ[0,2\pi ]$ is

• A. $0$
• B. $4$
• C. Infinite
• D. $1$

Answer 1

The correct option is D $1$

When taken the extreme cases with either sine or cosine becoming 1, we have LHS = $1$ and RHS $=3-1-0=2$ which don't match.
Also, when taken the middle case when x = 45^o, we have LHS =$\left[\sqrt{2}\right]=1$
RHS = $3+[-\frac{1}{\sqrt{2}}]+[-\frac{1}{\sqrt{2}}]=3-1-1=1$
Thus, we have one solution.