[latex]sin alpha = frac{12}{13} \ \ sin^{2} alpha +cos^{2} alpha =1 \ \ ( frac{12}{13})^{2} +cos^{2} alpha =1 \ \ cos^{2} alpha =1 - frac{144}{169} \ \ cos^{2} alpha = frac{25}{169} \ \ cos alpha = frac{5}{13} \ \ \ tg alpha = frac{sin alpha }{cos alpha }= frac{ frac{12}{13} }{ frac{5}{13} }= frac{12}{5} \ \ ctg alpha = frac{1}{tg alpha }= frac{5}{12} [/latex] [latex]tg^{2} alpha -8=0 \ \ tg^{2} alpha=8 \ \ tg alpha = sqrt{8}=2 sqrt{2} \ alpha -ostry \ tg alpha extgreater 0 \ \ \ left { {{tg alpha =2 sqrt{2}} atop {sin^{2}+cos^{2}=1}} ight.\ \ left { {{ frac{sin alpha }{cos alpha } =2 sqrt{2}} atop {sin^{2}+cos^{2}=1}} ight. \ \ left { {{sin alpha =2 sqrt{2}cos alpha } atop {sin^{2}+cos^{2}=1}} ight. \ \ (2 sqrt{2}cos alpha)^{2}+cos^{2} alpha =1 \ \ 8cos^{2} alpha +cos^{2} alpha =1 \ \ 9cos^{2} alpha =1 [/latex] [latex]cos^{2} alpha = frac{1}{9} \ \ cos alpha = frac{1}{3} \ \ cos alpha extgreater 0 \ \ sin alpha =2 sqrt{2}*cos alpha = 2 sqrt{2}* frac{1}{3}= frac{2 sqrt{2} }{3} [/latex]
