matematyka-funkcje trygonometryczne. zaadanie 18 - 19 w załączniku

18 [latex](sinalpha+cosalpha)(sinalpha-cosalpha)=frac{1}{3}[/latex] [latex]sin^2alpha-cos^2alpha=frac{1}{3}[/latex] [latex]1-cos^2alpha-cos^2alpha=frac{1}{3}[/latex] [latex]-cos^2alpha-cos^2alpha=frac{1}{3}-1[/latex] [latex]-2cos^2alpha=-frac{2}{3} /:(-2)[/latex] [latex]cos^2alpha=frac{1}{3}[/latex] [latex]cosalpha=sqrt{frac{1}{3}}[/latex] [latex]cosalpha=frac{ sqrt{3} }{3}[/latex] =============== 19 a) [latex]frac{3sinalpha-2cosalpha}{5cosalpha}=[/latex] [latex]frac{3sinalpha}{5cosalpha}+frac{2cosalpha}{5cosalpha}=[/latex] [latex]frac{3}{5}tgalpha+frac{2}{5}=frac{3}{5} cdot 3+frac{2}{5}=[/latex] [latex]frac{9}{5}+frac{2}{5}= frac{11}{5}=2 frac{1}{5}[/latex] b) [latex]frac{4sin^2alpha+5cos^2alpha}{sinalpha cosalpha}=[/latex] [latex]frac{4sin^2alpha}{sinalpha cosalpha}+frac{5cos^2alpha}{sinalpha cosalpha}=[/latex] [latex]frac{4sinalpha}{cosalpha}+frac{5cosalpha}{sinalpha}=[/latex] [latex]4tgalpha+ frac{5}{tgalpha} =4 cdot 3+ frac{5}{3} =12+1 frac{2}{3} =13 frac{2}{3}[/latex] c) [latex]frac{(sinalpha+cosalpha)^2}{cos^2alpha}=[/latex] [latex]frac{sin^2alpha+2sinalpha cosalpha+cos^2alpha}{cos^2alpha}=[/latex] [latex]frac{sin^2alpha}{cos^2alpha}+frac{2sinalpha cosalpha}{cos^2alpha}+frac{cos^2alpha}{cos^2alpha}=[/latex] [latex]tg^2alpha+frac{2sinalpha}{cosalpha}+1=[/latex] [latex]tg^2alpha+2tgalpha+1=[/latex] [latex]3^2+2cdot 3+1=9+6+1=16[/latex]