$\cos \alpha =\frac{1}{4}$  

 

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

${\sin }^2\alpha +{\left(\frac{1}{4}\right)}^2=1$  

${\sin }^2\alpha +\frac{1}{16}=1$  

${\sin }^2\alpha =\frac{15}{16}$  

$\sin \alpha =\frac{\sqrt{15}}{4}$  

 

$\mathrm{tg}\alpha =\frac{\frac{\sqrt{15}}{4}}{\frac{1}{4}}=\frac{\sqrt{15}}{4}\cdot \frac{4}{1}=\sqrt{15}$         

 

$2\sin \alpha +\frac{1}{2}\mathrm{tg}\alpha =\sout{2}{}^1\cdot \frac{\sqrt{15}}{\sout{4}{}_2}+\frac{1}{2}\cdot \sqrt{15}=\frac{1}{2}\sqrt{15}+\frac{1}{2}\sqrt{5}=\sqrt{15}$