# Simplify sin(cos−1x)

Francisca Rodden

## Answered question

2021-12-13

Simplify $\mathrm{sin}\left({\mathrm{cos}}^{-1}x\right)$

### Answer & Explanation

$\mathrm{cos}\left(a\right)=x=\frac{x}{1}$

$\mathrm{sin}\left({\mathrm{cos}}^{-1}x\right)=\mathrm{sin}\left(a\right)=\frac{\sqrt{1-{x}^{2}}}{1}=\sqrt{1-{x}^{2}}$ (using the Pythagorean Theorem)

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

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