Question

Calculating the principal value, find the value of $\cos \left[{\sin }^{-1}\left(\frac{3}{5}\right)\right]$.

Answer 1

We are to find $cos\left[si{n}^{-1}\left(\frac{3}{5}\right)\right]$

Let $si{n}^{-1}\left(\frac{3}{5}\right)=\theta$

where $\theta ϵ[-\frac{\pi }{2},\frac{\pi }{2}]$

So, $sin\theta =\frac{3}{5}$

so, $=cos\left[si{n}^{-1}\left(\frac{3}{5}\right)\right]$

$=cos\theta$

$=\sqrt{1-si{n}^2\theta }$

$=\sqrt{1-{\left(\frac{3}{5}\right)}^2}$

$=\sqrt{1-\frac{9}{25}}$

$=\sqrt{\frac{16}{25}}$

$=\frac{4}{5}$

So, $cos\left[si{n}^{-1}\left(\frac{3}{5}\right)\right]=\frac{4}{5}$