Question:
If $\sin ^{-1} x=\frac{\pi}{5}$ for some $x \in(-1,1)$, then the value of $\cos ^{-1} x$ is ____________________.
Solution:
Given: $\sin ^{-1} x=\frac{\pi}{5}, \forall x \in(-1,1)$
We know
$\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}$
$\Rightarrow \frac{\pi}{5}+\cos ^{-1} x=\frac{\pi}{2}$
$\Rightarrow \cos ^{-1} x=\frac{\pi}{2}-\frac{\pi}{5}=\frac{3 \pi}{10}$
Thus, the value of $\cos ^{-1} x$ is $\frac{3 \pi}{10}$.
If $\sin ^{-1} x=\frac{\pi}{5}$ for some $x \in(-1,1)$, then the value of $\cos ^{-1} x$ is $\frac{3 \pi}{10}$
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