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

Mar 9, 2017

$\cos \theta = \frac{\sqrt{8}}{3}$

#### Explanation:

$\sin \theta = \text{opp"/"hyp}$, in this case $\frac{1}{3}$.

Given these values, we can work out the $\text{adj}$ side of our imaginary triangle to work out $\cos \theta$ which equals $\text{adj"/"hyp}$.

"adj"=sqrt("hyp"^2-"opp"^2)-sqrt(3^2-1^2)=sqrt8

$\cos \theta = \text{adj"/"hyp} = \frac{\sqrt{8}}{3}$.

Another way we can work this out is using the trigonometric identity:

${\sin}^{2} A + {\cos}^{2} A \equiv 1$

$\cos A \equiv \sqrt{1 - {\sin}^{2} A}$

$\cos \theta = \sqrt{1 - {\left(\frac{1}{3}\right)}^{2}} = \sqrt{\frac{8}{9}} = \frac{\sqrt{8}}{3}$