# How do you find the exact value fo sin (cos^-1 (-1/4))

How do you find the exact value for $\mathrm{sin}\left({\mathrm{cos}}^{-1}\left(-\frac{1}{4}\right)\right)$?
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Zackary Galloway
${\mathrm{cos}}^{-1}\left(-\frac{1}{4}\right)$ is a t in $\left[0,\pi \right]$ with $\mathrm{cos}t=-\frac{1}{4}$
Since t is in quadrant 1 or 2, $\mathrm{sin}t$ is positive.
The problem becomes: $\mathrm{cos}t=-\frac{1}{4}$ and $\mathrm{sin}t>0$. Find $\mathrm{sin}t$.
Either draw a picture of t on the coordinate system,
or draw a right triangle with acute angle t, adjacent side 1 and hypotenuse 4
or use ${\mathrm{sin}}^{2}t+{\mathrm{cos}}^{2}=1$ to get
$\mathrm{sin}t=\frac{\sqrt{15}}{4}$