# If sin theta = sqrt5/3 then what is cos 2 theta?

Jun 11, 2016

$\setminus \cos \left(2 \setminus \theta\right) = - \frac{1}{9}$. It does not depend on the quadrant $\setminus \theta$ is in.

#### Explanation:

Double angle identities for cosine:

$\cos \left(2 \setminus \theta\right)$
$= 2 {\cos}^{2} \left(\setminus \theta\right) - 1$
$= {\cos}^{2} \left(\setminus \theta\right) - {\sin}^{2} \left(\setminus \theta\right)$
$= 1 - 2 {\sin}^{2} \left(\setminus \theta\right)$

Note that there are only even powers of $\sin \left(\setminus \theta\right)$ and $\cos \left(\setminus \theta\right)$. So any sign differences between quadrants cancels out, and if you know either $\sin \left(\setminus \theta\right)$ or $\cos \left(\setminus \theta\right)$ you have only one value for $\cos \left(2 \setminus \theta\right)$.