Question

Вычислите: 1) cos 2a, если cos a = 4/5 2) tg(a/2) , если cos a = -1/2 ответ : 1) 2)

Answer 1

$1)\ \cos \alpha =\frac{4}{5},\\\cos2 \alpha =2\cos^2 \alpha -1,\\\cos2 \alpha =2\bullet(\frac{4}{5})^2-1=2\bullet\frac{16}{25}-\frac{25}{25}=\frac{32-25}{25}=\frac{7}{25};\\\\2)\ \cos \alpha =-\frac{1}{2},\\tg\frac{ \alpha }{2}=\frac{\sin \alpha }{1+\cos \alpha }=\frac{\sqrt{1-\cos^2 \alpha }}{1+\cos \alpha },\\tg\frac{ \alpha }{2}=\frac{\sqrt{1-(-\frac{1}{2})^2}}{1-\frac{1}{2}}=\frac{\sqrt{1-\frac{1}{4}}}{\frac{1}{2}}=2\sqrt{\frac{3}{4}}=2\bullet\frac{\sqrt3}{2}=\sqrt3.$