Question

If ${\sin }^{-1}x=\frac{\pi }{5}$ for some$x\in [-1,1]$ then, find the value of ${\cos }^{-1}x.$

Answer 1

We have,

${\sin }^{-1}x=\frac{\pi }{5}$

We know that,

${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$

Now,

${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$

$\Rightarrow \frac{\pi }{5}+{\cos }^{-1}x=\frac{\pi }{2}$

$\Rightarrow {\cos }^{-1}x=\frac{\pi }{2}-\frac{\pi }{5}$

$\Rightarrow {\cos }^{-1}x=\frac{5\pi -2\pi }{10}$

$\Rightarrow {\cos }^{-1}x=\frac{3\pi }{10}$

Hence, this is the answer.