Вычислите sin a/2 и cos a/2, если cosa=-1/18 и π<a<3π/2.

\[\cos a = - \frac{1}{18};\ \ \ \]

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

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

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

\[= \sqrt{\frac{1 - \left( - \frac{1}{18} \right)}{2}} = \sqrt{\frac{1 + \frac{1}{18}}{2}} =\]

\[= \sqrt{\frac{19}{36}} = \frac{\sqrt{19}}{6}\text{\ \ }\]

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

\[= \sqrt{\frac{1 + \left( - \frac{1}{18} \right)}{2}} = \sqrt{\frac{1 - \frac{1}{18}}{2}} =\]

\[= \sqrt{\frac{17}{36}} = \left| \frac{\sqrt{17}}{6} \right| = - \frac{\sqrt{17}}{6}\ \]