c)
[tex]tg\alpha=-\frac13\\tg\alpha=\frac{sin\alpha}{cos\alpha}\\tg^2\alpha=\frac{sin^2\alpha}{cos^2\alpha}=\frac{sin^2\alpha}{1-sin^2\alpha}\\(-\frac13)^2=\frac{sin^2\alpha}{1-sin^2\alpha} /*1-sin^2\alpha\\\frac19-\frac19sin^2\alpha=sin^2\alpha /+\frac19sin^2\alpha\\\frac19=\frac{10}9sin^2\alpha /*\frac9{10}\\\frac1{10}=sin^2\alpha\\sin\alpha=\frac{\sqrt{10}}{10}\\\\cos^2\alpha=1-\frac1{10}=\frac9{10}\\cos\alpha=-\frac{3}{\sqrt{10}}=-\frac{3\sqrt{10}}{10}\\ctg\alpha=-3[/tex]
(W drugiej cwiartce ukladu wspolrzednych (α∈(90°; 180°)), tylko sinus jest dodatni)
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a)
[tex]sin\alpha=0.8\\0.64+cos^2\alpha=1\\cos^2\alpha=0.36\\cos\alpha=-0.6\\\\tg\alpha=\frac{0.8}{-0.6}=\frac8{10}*(-\frac{10}6)=-\frac86=-\frac43\\ctg\alpha=-\frac34[/tex]
b)
[tex]cos\alpha=-\frac14\\sin^2\alpha+\frac1{16}=1 /-\frac1{16}\\sin^2\alpha=\frac{15}{16}\\sin\alpha=\frac{\sqrt{15}}{4}\\\\tg\alpha=\frac{\sqrt{15}}4*(-4)=-\sqrt{15}\\ctg\alpha=-\frac{1}{\sqrt{15}}=-\frac{\sqrt{15}}{15}[/tex]
d)
[tex]ctg\alpha=-7\\tg\alpha=-\frac17\\\frac1{49}=\frac{sin^2\alpha}{1-sin^2\alpha} /*(1-sin^2\alpha)\\\frac1{49}-\frac1{49}sin^2\alpha=sin^2\alpha /+\frac1{49}sin^2\alpha\\\frac1{49}=\frac{50}{49}sin^2\alpha /*\frac{49}{50}\\\frac1{50}=sin^2\alpha\\sin\alpha=\frac{1}{5\sqrt2}=\frac{\sqrt2}{10}\\\\\frac1{50}+cos^2\alpha=1 /-\frac1{50}\\cos^2\alpha=\frac{49}{50}\\cos\alpha=-\frac{7}{5\sqrt2}=-\frac{7\sqrt2}{10}[/tex]
