Question

Find linear expression of $\sin^{-1}(\sin x)$ for $x \in [\frac{2025\pi}{2},\frac{2027\pi}{2}]$

Answer 1

1013π - x

Answer 2

1. Use the formula $\sin^{-1}(\sin x)= (-1)^n (x-n\pi)$ for $x\in [n\pi-\frac{\pi}{2}, n\pi+\frac{\pi}{2}]$.

2. For $x\in \left[\frac{2025\pi}{2},\frac{2027\pi}{2}\right]$, set $n\pi-\frac{\pi}{2}=\frac{2025\pi}{2}$ to get $n=1013$.

3. Since $1013$ is odd, $(-1)^{1013}=-1$.

4. Thus, $\sin^{-1}(\sin x)= - (x-1013\pi)=1013\pi - x$.