Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 675

\[1)\ z = \left( \frac{1}{2}\left( \cos\frac{\pi}{12} + i\sin\frac{\pi}{12} \right) \right)^{- 3} =\]

\[= 2^{3}\left( \cos\left( - \frac{3\pi}{12} \right) + i\sin\left( - \frac{3\pi}{12} \right) \right) =\]

\[= 8\left( \cos\left( - \frac{\pi}{4} \right) + i\sin\left( - \frac{\pi}{4} \right) \right) =\]

\[= 8\left( \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i \right) = 4\sqrt{2} - 4\sqrt{2}i.\]

\[2)\ z = \left( \sqrt{3} - i \right)^{6} =\]

\[= \left( 2\left( \frac{\sqrt{3}}{2} - \frac{1}{2}i \right) \right)^{6} =\]

\[= \left( 2\left( \cos\frac{\pi}{6} + i\sin\frac{\pi}{6} \right) \right)^{6} =\]

\[= 2^{6}\left( \cos\frac{6\pi}{6} + i\sin\frac{6\pi}{6} \right) =\]

\[= 64\left( \cos\pi + i\sin\pi \right) =\]

\[= 64 \bullet ( - 1) = - 64.\]

\[3)\ z = \frac{1}{\left( \cos{12{^\circ}} + i\sin{12{^\circ}} \right)^{5}} =\]

\[= \left( \cos{12{^\circ}} + i\sin{12{^\circ}} \right)^{- 5} =\]

\[= \cos( - 5 \bullet 12{^\circ}) + i\sin( - 5 \bullet 12{^\circ}) =\]

\[= \cos( - 60{^\circ}) + i\sin( - 60{^\circ}) =\]

\[= \cos\left( - \frac{\pi}{3} \right) + i\sin\left( - \frac{\pi}{3} \right) =\]

\[= \frac{1}{2} - \frac{\sqrt{3}}{2}i.\]

\[4)\ z =\]

\[= \frac{\left( \cos\left( - \frac{\pi}{3} \right) + i\sin\left( - \frac{\pi}{3} \right) \right)\left( 1 + \sqrt{3}i \right)^{7}}{i} =\]

\[= \frac{\left( \cos\left( - \frac{\pi}{3} \right) + i\sin\left( - \frac{\pi}{3} \right) \right)\left( 2\left( \frac{1}{2} + \frac{\sqrt{3}}{2}i \right) \right)^{7}}{i} =\]

\[= \frac{\left( \cos\left( - \frac{\pi}{3} \right) + i\sin\left( - \frac{\pi}{3} \right) \right) \bullet 2^{7}\left( \cos\frac{\pi}{3} + i\sin\frac{\pi}{3} \right)^{7}}{\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}} =\]

\[= 128\left( \cos\left( - \frac{\pi}{3} + \frac{7\pi}{3} - \frac{\pi}{2} \right) + i\sin\left( - \frac{\pi}{3} + \frac{7\pi}{3} - \frac{\pi}{2} \right) \right) =\]

\[= 128\left( \cos\left( 2\pi - \frac{\pi}{2} \right) + i\sin\left( 2\pi - \frac{\pi}{2} \right) \right) =\]

\[= 128\left( \cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2} \right) =\]

\(= 128\left( 0 + i \bullet ( - 1) \right) = - 128i.\)