Question

Find the value of:
$\sin {}^{ - 1} (sin \frac{3x}{5} )$
​

Answer 1

★Solution:★

We know that,

$\sin {}^{ - 1} (sin \: x) = x$

Therefore,

$\sin {}^{ - 1} ( \sin \frac{3x}{5} ) = \frac{3x}{5}$

But 3x/5 ∉ [-1/2,1/2], which is the principal branch of sin –¹x

However

$sin( \frac{3x}{5} ) = sin {}^{ - 1} (sin \frac{2x}{5})$

and 2x ∈ [-1/2,1/2]

Therefore,

$sin {}^{ - 1} (sin \frac{3x}{5} ) = sin {}^{ - 1} (sin \frac{2x}{5} ) = \frac{2x}{5}$

Hope it helps!!