$a)$  

$\cos \alpha =\frac{8}{10}=\frac{4}{5}$  

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

${\sin }^2\alpha =\frac{9}{25}$   

$\sin \alpha =\frac{3}{5}\vee \sin \alpha =-\frac{3}{5}$  

$\text{Wartosˊci funkcji trygonometrycznych kąta ostrego są dodatnie.}$  

$\sin \alpha =\frac{3}{5}$  

$\mathrm{tg}\alpha =\frac{\sin \alpha }{\cos \alpha }=\frac{\frac{3}{5}}{\frac{4}{5}}=\frac{3}{4}$

$\mathrm{ctg}\alpha =\frac{1}{\mathrm{tg}\alpha }=\frac{4}{3}$

 

$b)$  

$\cos \alpha =\frac{3}{5}$  

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

${\sin }^2\alpha =\frac{16}{25}$

$\sin \alpha =\frac{4}{5}$

$\mathrm{tg}\alpha =\frac{\frac{4}{5}}{\frac{3}{5}}=\frac{4}{3}$  

$\mathrm{ctg}\alpha =\frac{3}{4}$  

 

$c)$  

$\sin \alpha =\frac{1}{3}$  

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

${\cos }^2\alpha =\frac{8}{9}$  

$\cos \alpha =\frac{2\sqrt{2}}{3}$  

$\mathrm{tg}\alpha =\frac{\sin \alpha }{\cos \alpha }=\frac{\frac{1}{3}}{\frac{2\sqrt{2}}{3}}=\frac{1}{2\sqrt{2}}=\frac{\sqrt{2}}{4}$  

$\mathrm{ctg}\alpha =\frac{4}{\sqrt{2}}=2\sqrt{2}$  

 

$d)$  

$\sin \alpha =0,4=\frac{2}{5}$   

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

${\cos }^2\alpha =\frac{21}{25}$

$\cos \alpha =\frac{\sqrt{21}}{5}$  

$\mathrm{tg}\alpha =\frac{\sin \alpha }{\cos \alpha }=\frac{\frac{2}{5}}{\frac{\sqrt{21}}{5}}=\frac{2}{\sqrt{21}}=\frac{2\sqrt{21}}{21}$  

$\mathrm{ctg}\alpha =\frac{21}{2\sqrt{21}}=\frac{21\sqrt{21}}{42}=\frac{\sqrt{21}}{2}$

 

$e)$  

$\cos \alpha =\frac{\sqrt{13}}{7}$  

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

${\sin }^2\alpha =1-\frac{13}{49}=\frac{36}{49}$  

$\sin \alpha =\frac{6}{7}$  

$\mathrm{tg}\alpha =\frac{\sin \alpha }{\cos \alpha }=\frac{\frac{6}{7}}{\frac{\sqrt{13}}{7}}=\frac{6}{\sqrt{13}}=\frac{6\sqrt{13}}{13}$  

$\mathrm{ctg}\alpha =\frac{13}{6\sqrt{13}}=\frac{13\sqrt{13}}{78}=\frac{\sqrt{13}}{6}$   

 

$f)$  

$\mathrm{tg}\alpha =\frac{3}{2}$  

$b=3$  

$a=2$  

$c=\sqrt{a^2+b^2}=\sqrt{9+4}=\sqrt{13}$  

$\sin \alpha =\frac{3}{\sqrt{13}}=\frac{3\sqrt{13}}{13}$  

$\cos \alpha =\frac{2}{\sqrt{13}}=\frac{2\sqrt{13}}{13}$  

$\mathrm{ctg}\alpha =\frac{1}{\mathrm{tg}\alpha }=\frac{2}{3}$  

 

$g)$  

$\mathrm{tg}\alpha =2,4=\frac{24}{10}=\frac{12}{5}$  

$b=12$  

$a=5$  

$c=\sqrt{144+25}=\sqrt{169}=13$  

 

$\sin \alpha =\frac{12}{13}$  

$\cos \alpha =\frac{5}{13}$     

$\mathrm{ctg}\alpha =\frac{1}{\mathrm{tg}\alpha }=\frac{5}{12}$

 

$h)$

$\mathrm{ctg}\alpha =\frac{4}{5}$

$a=4$

$b=5$

$c=\sqrt{16+25}=\sqrt{41}$

 

$\sin \alpha =\frac{5}{\sqrt{41}}=\frac{5\sqrt{41}}{41}$

$\cos \alpha =\frac{4}{\sqrt{41}}=\frac{4\sqrt{41}}{41}$

$\mathrm{tg}\alpha =\frac{5}{4}$