Solve the equation in interval $[0,\pi]:[\cos x+\sin x]=[\cos x]+[\sin x]$,where [.] is the greatest integer function.
How should i start this question,breaking it into intervals is difficult.Please guide me.
[Math] $[\cos x+\sin x]=[\cos x]+[\sin x]$,where [.] is the greatest integer function.
functions
Best Answer
Key facts: (1) For $0\lt x\lt\pi/2$, we have $\sin x+\cos x\gt 1$, so we do not have equality.
(2) For $\pi/2\lt x\le 3\pi/4$, we have $\lfloor \cos x\rfloor=-1$ but $\lfloor \cos x+\sin x\rfloor=0$.
(3) For $3\pi/4\lt x\le \pi$ we have $\lfloor \cos x\rfloor=-1$ and $\lfloor \cos x+\sin x\rfloor=-1$ so the equality holds.
It remains to check $0$ and $\pi/2$.