Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 634

\[1)\ \left( 2\left( \cos\frac{4\pi}{3} + i\sin\frac{4\pi}{3} \right) \right)^{3} =\]

\[= 2^{3}\left( \cos\left( 3 \bullet \frac{4\pi}{3} \right) + i\sin\left( 3 \bullet \frac{4\pi}{3} \right) \right) =\]

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

\[= 8(1 + i \bullet 0) = 8;\]

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

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

\[= \cos\pi + i\sin\pi =\]

\[= - 1 + i \bullet 0 = - 1;\]

\[3)\ \left( \frac{1}{2}\left( \cos\frac{\pi}{8} + i\sin\frac{\pi}{8} \right) \right)^{4} =\]

\[= \left( \frac{1}{2} \right)^{4} \bullet \left( \cos\left( 4 \bullet \frac{\pi}{8} \right) + i\sin\left( 4 \bullet \frac{\pi}{8} \right) \right) =\]

\[= \frac{1}{16}\left( \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} \right) =\]

\[= \frac{1}{16} \bullet (0 + i \bullet 1) = \frac{1}{16}i;\]

\[4)\ \left( \cos\left( - \frac{\pi}{6} \right) + i\sin\left( - \frac{\pi}{6} \right) \right)^{9} =\]

\[= \cos\left( - \frac{\pi}{6} \bullet 9 \right) + i\sin\left( - \frac{\pi}{6} \bullet 9 \right) =\]

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

\[= 0 + i \bullet 1 = i.\]