oblicz wartość wyrażenia -3cos alfa +6sin alfa podzielić przez 8 sin alfa +3 cos alfa jeśli wiadomo że ctg alfa =-2

Rozwiązanie w załączniku.

[latex]ctgalpha=-2[/latex] [latex]frac{cosalpha}{sinalpha}=-2[/latex] [latex]cosalpha=-2sinalpha[/latex] [latex]sin^{2}alpha+cos^{2}alpha=1[/latex] [latex]sin^{2}alpha+4sin^{2}alpha=1[/latex] [latex]5sin^{2}alpha=1[/latex] [latex]sin^{2}alpha=frac{1}{5}[/latex] [latex]sinalpha=frac{sqrt{5}}{5} vee sinalpha=-frac{sqrt{5}}{5}[/latex] [latex]cosalpha=-2cdot frac{sqrt{5}}{5}=-frac{2sqrt{5}}{5} vee cosalpha=-2cdot (-frac{sqrt{5}}{5})=frac{2sqrt{5}}{5}[/latex] dla [latex] left { {{sinalpha=frac{sqrt{5}}{5}} atop {cosalpha=-frac{2sqrt{5}}{5}}} ight. [/latex] [latex]frac{-3cosalpha+6sinalpha}{8sinalpha+3cosalpha}=frac{frac{6sqrt{5}}{5}+frac{6sqrt{5}}{5}}{frac{8sqrt{5}}{5}-frac{6sqrt{5}}{5}}=frac{frac{12sqrt{5}}{5}}{frac{2sqrt{5}}{5}}=frac{12sqrt{5}}{5}cdotfrac{5}{2sqrt{5}}=6[/latex] dla [latex] left { {{sinalpha=-frac{sqrt{5}}{5}} atop {cosalpha=frac{2sqrt{5}}{5}}} ight. [/latex] [latex]frac{-3cosalpha+6sinalpha}{8sinalpha+3cosalpha}=frac{-frac{6sqrt{5}}{5}-frac{6sqrt{5}}{5}}{-frac{8sqrt{5}}{5}+frac{6sqrt{5}}{5}}=frac{-frac{12sqrt{5}}{5}}{-frac{2sqrt{5}}{5}}=6[/latex] [latex]frac{-3cosalpha+6sinalpha}{8sinalpha+3cosalpha}=6[/latex]