1. Wiedząc, że α jest kątem ostrym dla którego sin α = 2/3 oblicz wartość wyrażenia (cos α + tg α)² 2. Wiedząc, że sin α + cos α = 1/√2 oblicz sin α × cos α. 3. Wiadomo, że tg²α = 3 oraz α jest kątem ostrym. Podaj miarę kąta α.

1. [latex]sin alpha= frac{2}{3}[/latex] [latex](cos alpha + tg alpha )^2=cos^2alpha+2cos alpha tg alpha+tg^2 alpha=[/latex] [latex]1-sin^2alpha+2cos alpha frac{sinalpha}{cosalpha} +frac{sin^2alpha}{cos^2alpha}=[/latex] [latex]1-sin^2alpha+2 sinalpha+frac{sin^2alpha}{1-sin^2alpha}[/latex] --------------- [latex]1-left( frac{2}{3} ight)^2+2cdotfrac{2}{3}+ frac{left(frac{2}{3} ight)^2}{1-left( frac{2}{3} ight) ^2}=[/latex] [latex]1-frac{4}{9}+frac{4}{3}+ frac{frac{4}{9}}{1-frac{4}{9}}=frac{5}{9}+frac{12}{9}+ frac{frac{4}{9}}{frac{5}{9}}=[/latex] [latex]frac{17}{9}+frac{4}{5}=1frac{8}{9}+frac{4}{5}=1frac{40}{45}+frac{36}{45}=1frac{76}{45}=2 frac{31}{45}[/latex] ===================== 2. [latex]sin alpha+cosalpha=frac{1}{2} /()^2[/latex] [latex]sin^2alpha+2 sin alpha cosalpha+cos^2alpha=frac{1}{4}[/latex] [latex]1+2 sin alpha cosalpha=frac{1}{4}[/latex] [latex]2 sin alpha cosalpha=frac{1}{4}-1[/latex] [latex]2 sin alpha cosalpha=-frac{3}{4} /:2[/latex] [latex]sin alpha cosalpha=-frac{3}{8}[/latex] ===================== 3. [latex]tg^2 alpha=3[/latex] [latex]tgalpha=sqrt{3}[/latex] [latex]alpha=60^o[/latex]