Oblicz wartości pozostałych funkcji trygonometrycznych kąta ostrego alfa jeśli a)sin alfa = 1/3 b) tg alfa =4/5

[latex]a) \ sin alpha =frac{1}{3}\ \ sin^ alpha +cos^2 alpha =1\ (frac{1}{3})^2+cos^2 alpha =1\ frac{1}{9}+cos^2 alpha =1\ cos^2 alpha =frac{8}{9}\ cos alpha =sqrt{frac{8}{9}}=frac{2sqrt{2}}{3}\ \ \ tg alpha =frac{sin alpha }{cos alpha }=frac{frac{1}{3}}{frac{2sqrt{2}}{3}}=frac{1}{3}cdot frac{3}{2sqrt{2}}=frac{1}{2sqrt{2}}=frac{sqrt{2}}{4}\ \ ctg alpha =frac{1}{tg alpha }=frac{1}{frac{1}{2sqrt{2}}}=2sqrt{2}[/latex] [latex]tg alpha =frac{4}{5}\ frac{sin alpha }{cos alpha }=frac{4}{5}\ sin alpha =frac{4}{5}cos alpha \ \ sin^2 alpha +cos^2 alpha =1\ (frac{4}{5}cos alpha )^2+cos^2 alpha =1\ frac{16}{25}cos^2 alpha +cos^2 alpha =1\ frac{41}{25}cos^2 alpha =1\ cos^2 alpha =frac{25}{41}\ cos alpha =sqrt{frac{25}{41}}=frac{5}{sqrt{41}}=frac{5sqrt{41}}{41}\ \ sin alpha =frac{4}{5}cdot frac{5sqrt{41}}{41}=frac{4sqrt{41}}{41}\ \ [/latex] [latex]ctg alpha =frac{1}{tg alpha }=frac{1}{frac{4}{5}}=frac{5}{4}[/latex]