# How do you find the exact values of costheta and sintheta when tantheta=1/2?

Nov 30, 2016

There are two possible solutions:

$\sin \theta = \frac{1}{\sqrt{5}} \text{ } \cos \theta = \frac{2}{\sqrt{5}}$

$\sin \theta = - \frac{1}{\sqrt{5}} \text{ } \cos \theta = - \frac{2}{\sqrt{5}}$

#### Explanation:

Pose $x = \tan \theta$ and $y = \cos \theta$

As $\tan \theta = \frac{1}{2}$

$\frac{x}{y} = \frac{1}{2}$

and from the fundamental trigonometric identity:

${x}^{2} + {y}^{2} = 1$

Substituting $y$ from the first equation:

$y = 2 x$

${x}^{2} + {\left(2 x\right)}^{2} = 1$

$5 {x}^{2} = 1$

$x = \pm \frac{1}{\sqrt{5}}$
$y = \pm \frac{2}{\sqrt{5}}$