# How do you find the value for sin(arccos(-1/3))?

Jun 11, 2015

Use calculator to evaluate $\theta = \arccos \left(- \frac{1}{3}\right)$, allow for $\theta + \pi$ (since $\cos \left(\theta\right) = \cos \left(\theta + \pi\right)$); use calculator to evaluate the two values of theta (within the $0$ to $2 \pi$ range.

#### Explanation:

Using caluclator
$\textcolor{w h i t e}{\text{XXXX}}$$\arccos \left(- \frac{1}{3}\right) = 1.910633$
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$i.e $\cos \left(1.910633\right) = - \frac{1}{3}$
$\textcolor{w h i t e}{\text{XXXX}}$note that
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$$\cos \left(1.910633 + \pi\right)$ also $= - \frac{1}{3}$

Using calculator evaluate:
$\textcolor{w h i t e}{\text{XXXX}}$$\sin \left(1.910633\right) = 0.942809$
and
$\textcolor{w h i t e}{\text{XXXX}}$$\sin \left(1.910633 + \pi\right) = \sin \left(4.372552\right) = - 0.942809$

Jun 11, 2015

Find$\sin \left(\arccos \left(- \frac{1}{3}\right)\right)$

#### Explanation:

$\cos x = - \frac{1}{3}$--> arc x?

Calculator gives $x = \pm 109.47$.
Now, find $\sin \left(\pm 109.47\right)$

sin (109.47) = 0.94
sin (-109.47) = -0.94