Question:
Mark the tick against the correct answer in the following:
The value of $\sin \left(\cos ^{-1} \frac{3}{5}\right)$ is
A. $\frac{2}{5}$
B. $\frac{4}{5}$
C. $\frac{-2}{5}$
D. none of these
Solution:
To Find: The value of $\sin \left(\cos ^{-1} \frac{3}{5}\right)$
Now, let $x=\cos ^{-1} \frac{3}{5}$
$\Rightarrow \cos x=\frac{3}{5}$
Now, $\sin x=\sqrt{1-\cos ^{2} x}$
$=\sqrt{1-\left(\frac{3}{5}\right)^{2}}$
$=\frac{4}{5}$
$\Rightarrow x=\sin ^{-1} \frac{4}{5}=\cos ^{-1} \frac{3}{5}$
Therefore,
$\sin \left(\cos ^{-1} \frac{3}{5}\right)=\sin \left(\sin ^{-1} \frac{4}{5}\right)$
Let, $Y=\sin \left(\sin ^{-1} \frac{4}{5}\right)$
$\Rightarrow \sin ^{-1} \mathrm{Y}=\sin ^{-1} \frac{4}{5}$
$\Rightarrow \mathrm{Y}=\frac{4}{5}$