[latex]a)\ sin alpha =0,8=frac{8}{10}=frac{4}{5}\ \ sin^2 alpha +cos^2 alpha =1\ (frac{4}{5})^2+cos^2 alpha =1\ frac{16}{25}+cos^2 alpha =1\ cos^2 alpha =frac{9}{25}\ cos alpha =sqrt{frac{9}{25}}+frac{3}{5}\ \ tg alpha =frac{sin alpha }{cos alpha }=frac{frac{4}{5}}{frac{3}{5}}=frac{4}{3}\ \ ctg alpha =frac{1}{tg alpha }=frac{1}{frac{4}{3}}=frac{3}{4}[/latex] [latex]b)\ ctg alpha = frac{cos alpha }{sin alpha }=2sqrt{3}\ cos alpha =2sqrt{3}sin alpha \ \ sin^2 alpha +cos^2 alpha =1\ sin^2 alpha +(2sqrt{3}sin alpha )^2=1\ sin^2 alpha +12sin^2 alpha =1\ 13sin^2 alpha =1\ sin^2 alpha =frac{1}{13}\ sin alpha =sqrt{frac{1}{13}}=frac{sqrt{13}}{13}\ \ cos alpha =2sqrt{3}cdot frac{sqrt{13}}{13}=frac{2sqrt{39}}{13}\ \ tg alpha =frac{1}{ctg alpha }=frac{1}{2sqrt{3}}=frac{sqrt{3}}{6} [/latex] [latex]c)\ tg alpha =frac{sin alpha }{cos alpha }=frac{3}{2}\ sin alpha =frac{3}{2}cos alpha \ \ ctg alpha =frac{1}{tg alpha }=frac{1}{frac{3}{2}}=frac{2}{3}\ \ sin^2 alpha +cos^2 alpha =1\ (frac{3}{2}cos alpha )^2+cos^2 alpha =1\ frac{9}{4}cos^2 alpha +cos^2 alpha =1\ frac{13}{4}cos^2 alpha =1\ cos^2 alpha =frac{4}{13}\ cos alpha =sqrt{frac{4}{13}}=frac{2}{sqrt{13}}=frac{2sqrt{13}}{13}\ \ sin alpha =frac{3}{2}cdot frac{2sqrt{13}}{13}=frac{3sqrt{13}}{13}[/latex]
