Mathematics

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


SOLUTION
$$\sin^{-1}\left ( \cos x \right )= x+\frac{\Pi}{2}$$
Explanation:
We know that the cosine function, is nothing more than the $$sine \frac{\Pi }{2} $$ radians out of phase, as proved below:
$$ cos(θ−\frac{\Pi }{2}) = cos(θ)cos(−\frac{\Pi }{2})−sin(θ)sin(−\frac{\Pi }{2})$$
$$ cos(θ−\frac{\Pi }{2}) = cos(θ). 0 −\left (-sin(θ)sin(\frac{\Pi }{2})\right )$$
$$ cos(θ−\frac{\Pi }{2}) = \sin(θ) $$
So we can say that the sine function, 90 degrees ahead, is the cosine function.
Using the property of inverse functions that \displaystyle $$ f^{-1}\left ( f\left ( x \right ) \right )=x $$
$$\sin^{-1}\left ( \cos x \right )= x+\frac{\Pi}{2}$$
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Subjective Medium Published on 17th 09, 2020
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