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

 

Z jedynki trygonometrycznej:

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

${\sin }^2\alpha +{\left(\frac{1}{2}\sin \alpha \right)}^2=1$  

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

$\frac{5}{4}{\sin }^2\alpha =1$  

${\sin }^2\alpha =\frac{4}{5}$  

$\left|\sin \alpha \right|=\frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5}$   

Kąt jest ostry a więc sinus jest dodatni:

$\sin \alpha =\frac{2\sqrt{5}}{5}$  

 

A więc:

$\cos \alpha =\frac{1}{2}\sin \alpha =\frac{1}{2}\cdot \frac{2\sqrt{5}}{5}=\frac{\sqrt{5}}{5}$