Question

1)Найти $sin alpha$ и $tg alpha$, если $cos alpha =- frac{2}{5}$ и $pi textless alpha textless frac{3 pi }{2}$
2) Найти $cos alpha$ и $ctg alpha$, если $sin alpha = frac{3}{4}$
и $frac{ pi }{2} textless alpha textless pi$

Answer 1

$1); ; cos alpha =-frac{2}{5}; ,\\sin alpha =pm sqrt{1-cos^2 alpha }; ,; tg alpha =pm sqrt{frac{1}{cos^2 alpha }-1}\\; ; alpha in (pi ;frac{3pi}{2}); ; to ; ; sin alpha textless 0,; ; tg alpha textgreater 0\\sin alpha =-sqrt{1-frac{4}{25}}=-sqrt{frac{21}{25}}=-frac{sqrt{21}}{5}\\tg alpha =+sqrt{frac{25}{4}-1}=frac{sqrt{21}}{2}$

$2); ; sin alpha =frac{3}{4}\\ alpha in (frac{pi}{2};pi ); ; to ; ; cos alpha textless 0; ,; ; ctg alpha textless 0\\cos alpha =-sqrt{1-sin^2 alpha }=-sqrt{1-frac{9}{16}}=-frac{sqrt7}{4}\\ctg alpha =-sqrt{frac{1}{sin^2 alpha }-1}=-sqrt{frac{16}{9}-1}=-frac{sqrt7}{3}$