Value of cos sin inverse x and sin cos inverse x

In this post, we will cover , how we can convert cos sin inverse x and sin cos inverse x in the terms of x

Value of cos sin inverse x

$f(x) =cos sin^{-1} x$

Let $\theta = sin^{-1} x$
or $sin \theta =x$

Now we know that $\theta \in [-\pi/2, \pi/2]$
Also cos is positive in Ist and IV quadrant.

Therefore $cos \theta$ is positive. it will give by

$cos \theta = \sqrt {1 – sin^2 \theta } =\sqrt {1 -x^2}$

Therefore

$f(x) =cos sin^{-1} x = \sqrt {1 -x^2}$

Value of sin cos inverse x

$f(x) =sin cos^{-1} x$

Let $\theta = cos^{-1} x$
or $cos \theta =x$

Now we know that $\theta \in [0, \pi]$
Also sin is positive in Ist and II quadrant.

Therefore $sin \theta$ is positive. it will give by

$sin \theta = \sqrt {1 – cos^2 \theta } =\sqrt {1 -x^2}$

Therefore

$f(x) =sin cos^{-1} x = \sqrt {1 -x^2}$

Summary

$cos sin^{-1} x = sin cos ^{-1} x=\sqrt {1 -x^2}$