Zadanie 12
a)
tg\alpha=\frac{1}{3} , 180^{\circ} < \alpha < 270^{\circ} , III ćwiartka sinus, cosinus ujemne
\frac{sin\alpha}{cos\alpha}=\frac{1}{3}
cos\alpha=3sin\alpha
sin^2\alpha+cos^2\alpha=1
sin^2\alpha+(3sin\alpha)^2=1
sin^2\alpha+9sin^2\alpha=1
10sin^2\alpha=1 \ |:10
sin^2\alpha=\frac{1}{10}
sin\alpha =-\sqrt{\frac{1}{10}}
sin\alpha=-\frac{1}{\sqrt{10}}=-\frac{1}{\sqrt{10}}\cdot \frac{\sqrt{10}}{\sqrt{10}}
sin\alpha=-\frac{\sqrt{10}}{10}
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cos\alpha=3sin\alpha=-\frac{3\sqrt{10}}{10}
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ctg\alpha = \frac{1}{tg\alpha}=\frac{1}{\frac{1}{3}}=3
b)
sin\alpha=\frac{12}{13} , 90^{\circ} < \alpha < 180^{\circ} II ćwiartka tylko sinus dodatni
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sin^2\alpha+cos^2\alpha=1
(\frac{12}{13})^2+cos^2\alpha=1
cos^2\alpha=1-\frac{144}{169}
cos\alpha=-\sqrt{\frac{25}{169}}
cos\alpha =-\frac{5}{13}
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tg\alpha=\frac{sin\alpha}{cos\alpha}=\frac{\frac{12}{13}}{-\frac{5}{13}}=-\frac{12}{13}\cdot \frac{13}{5}=-\frac{12}{5}
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ctg\alpha=\frac{1}{tg\alpha}=\frac{1}{-\frac{12}{5}}=-\frac{5}{12}