Q:

Find the value of $\sin (x)$ if $\cos(x)=\frac 1 3$

$$\cos (x)=\left (\frac{1}{3}\right)$$for $0°\lt\ x\lt90°$
Find the exact value of $\sin(x)$.
Give your answer as a surd.
I found the value with my calculator
$$\sin (x)=\left(\frac{\sqrt 8}{3}\right)$$
Is there a way of doing this by hand? 

A:

$\sin^2 x + \cos^2 x = 1$
$\sin^2 x = 1 - \cos^2 x$
$\sin x = \pm \sqrt{1 - \cos^2 x}$
$\sin x = \pm \sqrt{1 - \left( \frac 13\right)^2}$
$\sin x = \pm \sqrt{1 - \frac 19}$
$\sin x = \pm \sqrt{\frac 89}$
$\sin x = \pm \frac {\sqrt 8}3$

