Planimetria 1. Wiedząc, że cosL=7/8 oblicz sin,tg,ctg. 2. Oblicz wartość wyrażenia frac{tg30-sin30}{cos60} 3. Oblicz 1- frac{sin42}{cos48} 4. Wiedząc, że przyprostokątna (b)=4 a przeciwprostokątna= 10 w trójkącie prostokątnym, oblicz tg i ctg. 5.W

1. [latex]cos alpha = frac{7}{8} \\ sin^2 alpha + cos^2 alpha = 1 \\ sin^2 alpha + (frac{7}{8})^2= 1 \\ sin^2 alpha + frac{49}{64}= 1 \\ sin^2 alpha =1- frac{49}{64} \\ sin^2 alpha = frac{64}{64} - frac{49}{64} \\ sin^2 alpha =frac{15}{64} \\ sin alpha =sqrt{frac{15}{64}} vee sin alpha =-sqrt{frac{15}{64}} \\ oxed{sin alpha =frac{sqrt{15}}{8} vee sin alpha =-frac{sqrt{15}}{8}}[/latex] [latex]tg alpha = frac{sin alpha}{cosalpha} \\ tg alpha =frac{sqrt{15}}{8} : frac{7}{8} vee tg alpha =-frac{sqrt{15}}{8} : frac{7}{8} \\ tg alpha =frac{sqrt{15}}{8} cdot frac{8}{7} vee tg alpha =-frac{sqrt{15}}{8} cdot frac{8}{7} \\ oxed{tg alpha =frac{sqrt{15}}{7} vee tg alpha =-frac{sqrt{15}}{7}}[/latex] [latex]ctg alpha = frac{cos alpha}{sinalpha}= frac{1}{tg alpha} \\ ctg alpha =1 : frac{sqrt{15}}{7} vee ctg alpha =1 : (- frac{sqrt{15}}{7})\\ ctg alpha =1 cdot frac{7}{sqrt{15}} vee ctg alpha =1 cdot (- frac{7}{sqrt{15}})\\ ctg alpha =frac{7}{sqrt{15}} vee ctg alpha =- frac{7}{sqrt{15}}[/latex] [latex]ctg alpha =frac{7 cdot sqrt{15}}{sqrt{15} cdot sqrt{15}} vee ctg alpha =- frac{7 cdot sqrt{15}}{sqrt{15} cdot sqrt{15}} \\ oxed{ctg alpha =frac{7sqrt{15}}{15} vee ctg alpha =-frac{7sqrt{15}}{15}}[/latex] 2. [latex] frac{tg30^o-sin30^o}{cos60^o} = ( frac{sqrt{3}}{3} - frac{1}{2}) : frac{1}{2} = ( frac{2sqrt{3}}{6} - frac{3}{6}) cdot frac{2}{1} =frac{2sqrt{3}-3}{6} cdot frac{2}{1} = \\ = frac{2sqrt{3}-3}{3}= frac{2sqrt{3}}{3}-frac{3}{3}= oxed{frac{2sqrt{3}}{3}-1}[/latex] 3. [latex]1- frac{sin42^o}{cos48^o} =1- frac{sin(90^o-48^o)}{cos48^o} =1- frac{cos48^o}{cos48^o} =1-1=0[/latex] Dla zad.4 i 5 oznaczenia jak na rysunku - patrz załącznik 4. b = 4; c = 10 Drugą przyprostokątną a obliczymy z tw. Pitagorasa: [latex]a^2 + b^2 = c^2 \\ a^2 +4^2 = 10^2 \\ a^2+16 = 100 \\ a^2 = 100 - 16 \\ a^2 = 84 \\ a = sqrt{84} = sqrt{4 cdot 21}= 2sqrt{21}[/latex] [latex]a=2sqrt{21} i b = 4 \\ tg alpha = frac{a}{b} = frac{2sqrt{21}}{4} = frac{sqrt{21}}{2} \\ ctg alpha= frac{b}{a} = frac{4}{2sqrt{21}} = frac{2}{sqrt{21}} = frac{2 cdot sqrt{21}}{sqrt{21} cdot sqrt{21}} = frac{2sqrt{21}}{21} \\ tg eta= frac{b}{a} = frac{2sqrt{21}}{21} \\ ctg eta= frac{a}{b} = frac{sqrt{21}}{2}[/latex] 5. [latex]b = 12; cosalpha = frac{3}{5}; c =x \\ cosalpha = frac{b}{c} \\ frac{3}{5} =frac{12}{x} \\ 3 cdot x = 5 cdot 12 \\ 3x = 60 /:3 \\ oxed{x = 20}[/latex] lub [latex]b = 12; coseta = frac{3}{5}; c =x \\ coseta= frac{a}{c} \\ frac{3}{5} =frac{a}{x} \\ 5 cdot a = 3 cdot x \\ 5a = 3x /:5 \\ a= frac{3x}{5} \\ Z tw. Pitagorasa: \\ c^2 = a^2 + b^2 \\ x^2 = (frac{3x}{5})^2 + 12^2 \\ x^2 = frac{9x^2}{25}+ 144 / cdot 25 \\ 25x^2 = 9x^2 +3600 \\ 25x^2 - 9^2 = 3600 \\ 16x^2 = 3600 /:16 \\ x^2 = 225 \\ oxed{x = 15}[/latex]