Question

Найти sin2a если sina=1/2 и pi/2<a<pi Найти tg2a если sina=1/2 и pi/2<a<pi

Answer 1

$sina= \frac{1}{2}\\\\ \frac{\pi}{2}\ \textless \ a\ \textless \ \pi \; \; \; \to \; \; cosa\ \textless \ 0\; ,\\\\ cosa=-\sqrt{1-sin^2a}=-\sqrt{1-\frac{1}{4}}=-\frac{\sqrt3}{2}\\\\sin2a=2\cdot sina\cdot cosa=-2\cdot \frac{1}{2}\cdot \frac{\sqrt3}{2}=-\frac{\sqrt3}{2}\\\\ tg2a= \frac{sin2a}{cos2a}= \frac{sin2a}{cos^2a-sin^2a} = \frac{-\frac{\sqrt3}{2}}{\frac{3}{4}-\frac{1}{4}}=-\frac{\frac{\sqrt3}{2}}{\frac{2}{4}}=- \frac{\sqrt3\cdot 4}{2\cdot 2} =-\sqrt3$