If $a=\sin^{-1}(\sin5)$ and $b=\cos^{-1}(\cos5)$, then $a^2+b^2$ is equal to
Step-by-Step Solution
Key Concept: Reduce $\sin^{-1}(\sin5)$ using the fact that $5\in[3\pi/2,2\pi]$... actually $5\approx5$ rad $\in(3\pi/2,2\pi)$: $\sin^{-1}(\sin5)=5-2\pi$ (since $5\in(\pi,2\pi)$ and principal value). For $\cos^{-1}(\cos5)$: $5\in(\pi,2\pi)$, so $\cos^{-1}(\cos5)=2\pi-5$.
$a=\sin^{-1}(\sin5)=5-2\pi$, $b=\cos^{-1}(\cos5)=2\pi-5$. $a^2+b^2=(5-2\pi)^2+(2\pi-5)^2=8\pi^2-40\pi+50$.
Correct Answer: 2