# How do you use sintheta=1/3 to find tantheta?

Jan 8, 2017

$\pm \frac{\sqrt{2}}{4}$

#### Explanation:

First find cos t.
${\cos}^{2} t = 1 - {\sin}^{2} t$
${\cos}^{2} t = 1 - \frac{1}{9} = \frac{8}{9}$
$\cos t = \pm \frac{2 \sqrt{2}}{3}$
$\tan t = \sin \frac{t}{\cos t} = \pm \left(\frac{1}{3}\right) \left(\frac{3}{2 \sqrt{2}}\right) = \pm \frac{1}{2 \sqrt{2}} = \pm \frac{\sqrt{2}}{4}$
Note about the signs of tan t.
On the unit circle:
sin t = 1/3 --> cos t either + or - , there for --> tan t either + or -
Check by calculator.
$\sin t = \frac{1}{3}$ --> $t = {19}^{\circ} 47$ and $t = 180 - 19.47 = {160}^{\circ} 53$
tan 19.47 = 0.35 and tan 160.53 = - 0.35
$\pm \frac{\sqrt{2}}{4} = \pm \frac{1.414}{5} = \pm 0.35$ OK