# How do you simplify Sin(Cos^-1 x)?

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May 9, 2016

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

#### Explanation:

Let's draw a right triangle with an angle of $a = {\cos}^{- 1} \left(x\right)$.

As we know $\cos \left(a\right) = x = \frac{x}{1}$ we can label the adjacent leg as $x$ and the hypotenuse as $1$. The Pythagorean theorem then allows us to solve for the second leg as $\sqrt{1 - {x}^{2}}$.

With this, we can now find $\sin \left({\cos}^{- 1} \left(x\right)\right)$ as the quotient of the opposite leg and the hypotenuse.

$\sin \left({\cos}^{- 1} \left(x\right)\right) = \sin \left(a\right) = \frac{\sqrt{1 - {x}^{2}}}{1} = \sqrt{1 - {x}^{2}}$

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