# How do you simplify sin(arccos(x))?

Oct 21, 2016

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

#### Explanation:

From Pythagoras, we have:

${\sin}^{2} \theta + {\cos}^{2} \theta = 1$

If $x \in \left[- 1 , 1\right]$ and $\theta = \arccos \left(x\right)$ then:

$\theta \in \left[0 , \pi\right]$

$\sin \left(\theta\right) \ge 0$

Hence:

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

Note we can use the non-negative square root since we have already established that $\sin \left(\arccos \left(x\right)\right) \ge 0$