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# How do you determine tantheta given sintheta=-1/3, 180^circ<theta<270^circ?

Dec 16, 2016

$\tan \left(\theta\right) = \frac{1}{2 \sqrt{2}}$

#### Explanation:

If ${180}^{\circ} < \theta < {270}^{\circ}$
and $\sin \left(\theta\right) = - \frac{1}{3}$
then

The adjacent side is calculated using the Pythagorean Theorem as
$- \sqrt{{3}^{2} - {1}^{2}} = - 2 \sqrt{2}$
Givin
tan(theta)="opposite"/"adjacent"=((-1))/((-2sqrt(2)))= 1/(2sqrt(2)

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