Вычислите sin a/2 и cos a/2, если cosa=-1/8 и π<a<3π/2.

\[\cos a = - \frac{1}{8};\ \ \ \ \ \]

\[\pi < a < \frac{3\pi}{2} \Longrightarrow \frac{\pi}{2} < \frac{a}{2} < \frac{3\pi}{4} \Longrightarrow\]

\[\Longrightarrow \sin\frac{a}{2} > 0;\ \cos\frac{a}{2} < 0\]

\[\sin\frac{a}{2} = \sqrt{\frac{1 - \cos a}{2}} =\]

\[= \sqrt{\frac{1 - \left( - \frac{1}{8} \right)}{2}} = \sqrt{\frac{1 + \frac{1}{8}}{2}} =\]

\[= \sqrt{\frac{9}{16}} = \left| \frac{3}{4} \right| = \frac{3}{4}\]

\[\cos\frac{a}{2} = \sqrt{\frac{1 + \cos a}{2}} =\]

\[= \sqrt{\frac{1 + \left( - \frac{1}{8} \right)}{2}} = \sqrt{\frac{1 - \frac{1}{8}}{2}} =\]

\[= \sqrt{\frac{7}{16}} = \left| \frac{\sqrt{7}}{4} \right| = - \frac{\sqrt{7}}{4}\]