Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 509

\[\boxed{\mathbf{509}\mathbf{.}}\]

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

\[\left( \sin a + \cos a \right)^{2} = \left( \frac{1}{2} \right)^{2}\]

\[\sin^{2}a + \cos^{2}a + 2\sin a \bullet \cos a =\]

\[= \frac{1}{4}\]

\[1 + \sin{2a} = \frac{1}{4}\]

\[\sin{2a} = \frac{1}{4} - 1 = \frac{1}{4} - \frac{4}{4} = - \frac{3}{4}\]

\[Ответ:\ - \frac{3}{4}\]

\[2)\sin a - \cos a = - \frac{1}{3}\]

\[\left( \sin a - \cos a \right)^{2} = \left( - \frac{1}{3} \right)^{2}\]

\[\sin^{2}a + \cos^{2}a - 2\sin a \bullet \cos a =\]

\[= \frac{1}{9}\]

\[1 - \sin{2a} = \frac{1}{9}\]

\[\sin{2a} = 1 - \frac{1}{9} = \frac{9}{9} - \frac{1}{9} = \frac{8}{9}\]

\[Ответ:\ \ \frac{8}{9}.\]