# How do you find the exact value of cos(theta) if sin(theta)=-2/3?

Apr 2, 2018

cos $\theta$= $\frac{\sqrt{5}}{3}$
or it could be cos $\theta$= $\frac{\sqrt{5}}{-} 3$

#### Explanation:

Since sin $\theta$ is negative, it can be in the third or fourth quadrant

Drawing your right-angled triangle, place your $\theta$ in one of three corners. Your longest side will be 3 and the side opposite the $\theta$ will be -2. Finally, using Pythagoras theorem, your last side should be $\sqrt{5}$

Now, if your triangle was in the third quadrant, you would have
cos $\theta$= $\frac{\sqrt{5}}{-} 3$ since cosine is negative in the third quadrant

But if your triangle was in the fourth quadrant, you would have
cos $\theta$= $\frac{\sqrt{5}}{3}$ since cosine is positive in the fourth quadrant

Apr 2, 2018

color(indigo)(cos theta = +- sqrt5 / 3

#### Explanation:

$\sin \theta = - \frac{2}{3}$

${\sin}^{2} \theta = {\left(- \frac{2}{3}\right)}^{2} = \frac{4}{9}$

But ${\cos}^{2} \theta = 1 - {\sin}^{2} \theta = 1 - \frac{4}{9} = \frac{5}{9}$

$\therefore \cos \theta = \sqrt{\frac{5}{9}} = \pm \frac{\sqrt{5}}{3}$