# What is sin and cos if tan = 1/2 and sin >0?

Apr 14, 2015

$\sin \alpha = \frac{\sqrt{5}}{5}$,

$\cos \alpha = 2 \frac{\sqrt{5}}{5}$.

First of all, if the sinus and the tangent of an angle $\alpha$ is positive, the angle is in the first quadrant and so sinus and cosine are positive!

Since $\tan \alpha = \sin \frac{\alpha}{\cos} \alpha$,

than:

$\sin \frac{\alpha}{\cos} \alpha = \frac{1}{2}$

and, for the fundamental relation of trigonometry:

${\sin}^{2} \alpha + {\cos}^{2} \alpha = 1$,

let's solve the system!

$\cos \alpha = 2 \sin \alpha$

${\sin}^{2} \alpha + {\cos}^{2} \alpha = 1$

than

${\sin}^{2} \alpha + 4 {\sin}^{2} \alpha = 1 \Rightarrow {\sin}^{2} \alpha = \frac{1}{5} \Rightarrow \sin \alpha = \frac{\sqrt{5}}{5}$

(only the positive value!)

and

$\cos \alpha = 2 \sin \alpha = 2 \frac{\sqrt{5}}{5}$.