Wiedząc, że alfa jest kątem rozwartym oraz sinx+ cosx= 7/13, oblicz: a) sinx, cosx b) tgx+ ctgx

[latex]sin alpha +cos alpha =frac{7}{13}\ sin alpha = frac{7}{13}-cos alpha \ \ sin^2 alpha +cos^2 alpha =1\ [/latex] [latex](frac{7}{13}-cos alpha )^2+cos^2 alpha =1\ frac{49}{169} - frac{14}{13}cos alpha + cos^2 alpha +cos^2 alpha =1\ 2cos^2 alpha -frac{14}{13}cos alpha -frac{120}{169}=0\ Delta=(-frac{14}{13})^2-4cdot 2cdot (-frac{120}{169})=frac{196}{169}+frac{960}{169} = frac{1156}{169}\ sqrt{Delta} =sqrt{frac{1156}{169}}=\frac{34}{13}\ x=frac{frac{14}{13}-frac{34}{ 13} }{4}=frac{-frac{20}{13}}{4}=-frac{5}{13}\ \ [/latex] [latex]cos alpha =-frac{5}{13}\ \ sin^2 alpha =1-cos^2 alpha =1-frac{25}{169}=frac{144}{169}\ sin alpha =sqrt{frac{144}{169}}=frac{12}{13}\ \ b)\ 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]