1. a) [latex]sinalpha=frac{a}{c}\ coseta=frac{x}{r}\tggamma=frac{m}{k}\ctgdelta=frac{v}{t}[/latex] b) [latex]sindelta=frac{t}{z}\cosalpha=frac{b}{c}\tgeta=frac{y}{x}\ctggamma=frac{k}{m}[/latex] c) [latex]singamma=frac{m}{l}\cosdelta=frac{v}{z}\tgalpha=frac{a}{b}\ctgeta=frac{x}{y}[/latex] d) [latex]sineta=frac{y}{r}\cosgamma=frac{k}{l}\tgdelta=frac{t}{v}\ctgalpha=frac{b}{a}[/latex] 2. a) sin²45°+cos30°-tg60°-sin30°=[latex]2(frac{sqrt{2}}{2})^2[/latex] + [latex]frac{sqrt{3}}{2}[/latex] - 1 - [latex]frac{sqrt{3}}{2}[/latex] = ½-1 = -½ b) 2sin60°-4cos60°-tg60°-sin30°=2·[latex]frac{sqrt{3}}{2}[/latex] - [tex] sqrt{3} [/latex]- ½ = -2½ c) tg60°+4sin60°-2cos30°-3ctg60°=[latex]sqrt{3}[/latex] + 4 ·[latex]frac{sqrt{3}}{2}[/latex] - 2 ·[latex]frac{sqrt{3}}{2}[/latex] -3· [latex]frac{sqrt{3}}{2}[/latex] = [latex] sqrt{3} [/latex] 3. h-wysokość trójkata x=z+y [latex]cos60^o=frac{z}{5}\frac{1}{2}=frac{z}{5}\2z=5/:2\z=2,5[/latex] [latex]sin60^o=frac{h}{5}\frac{sqrt{3}}{2}=frac{h}{5}\5sqrt{3}=2h\h=2,5sqrt{3}[/latex] [latex]tg45^o=frac{h}{y}\1=frac{2,5sqrt{3}}{y}\y=2,5sqrt{3}[/latex] [latex]x=2,5+2,5 [/latex] + [latex] sqrt{3}[/latex] b) 45°+15°=60° y- dł boku (drugiej przyprostokątnej) w tym mniejszym trójkąciie prostokątnym [latex]tg45^o=frac{y}{16}\1=frac{y}{16}\y=16[/latex] [latex]tg60^o=frac{yplus x}{16}\sqrt{3}=frac{16plusx}{16}\16sqrt{3}=16plusx\x=16-16sqrt{3}[/latex] wyraz plus oz działanie + (w latex nie dnaiła symbol dodawania) 4 a) cosx·sinx·ctgx+sin²x= [latex]cosx cdot sinx cdot frac{cosx}{sinx}[/latex] + sin²x=cos²x+sin²x=1 b) ctg²x+1= [latex]frac{cos^2x}{sin^2x}[/latex] +1 = [latex]frac{cos^2x plus sin^2x}{sin^2x} = frac{1}{sin^2x}[/latex] bo sin²x+cos²x=1 c){sinx+tgx}/sinx - 1/cosx = [latex]frac{sinx}{sinx}[/latex] + [latex]frac{tgx}{sinx} - frac{1}{cosx} = 1[/latex] + [latex]frac{sinx}{cosx} cdot frac{1}{sinx} - frac{1}{cosx} = 1[/latex] + [latex]frac{1}{cosx}-frac{1}{cosx}=1[/latex]
