Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 674

\[1)\ \left( 3\left( \cos\frac{7\pi}{8} + i\sin\frac{7\pi}{8} \right) \right)^{4} =\]

\[= 3^{4}\left( \cos\left( 4 \bullet \frac{7\pi}{8} \right) + i\sin\left( 4 \bullet \frac{7\pi}{8} \right) \right) =\]

\[= 81\left( \cos\frac{7\pi}{2} + i\sin\frac{7\pi}{2} \right) =\]

\[= 81\left( \cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2} \right) =\]

\[= 81(0 - i) = - 81i.\]

\[2)\ \left( \cos{20{^\circ}} + i\sin{20{^\circ}} \right)^{12} =\]

\[= \cos(12 \bullet 20{^\circ}) + i\sin(12 \bullet 20{^\circ}) =\]

\[= \cos{240{^\circ}} + i\sin{240{^\circ}} =\]

\[= \cos\frac{4\pi}{3} + i\sin\frac{4\pi}{3} = - \frac{1}{2} - \frac{\sqrt{3}}{2}i.\]

\[3)\ \left( 2\left( \cos( - 20{^\circ}) + i\sin( - 20{^\circ}) \right) \right)^{3} =\]

\[= 2^{3}\left( \cos( - 20{^\circ} \bullet 3) + i\sin( - 20{^\circ} \bullet 3) \right) =\]

\[= 8\left( \cos( - 60{^\circ}) + i\sin( - 60{^\circ}) \right) =\]

\[= 8\left( \frac{1}{2} - \frac{\sqrt{3}}{2}i \right) = 4 - 4\sqrt{3}i.\]

\[4)\ \frac{1}{\left( \cos\frac{\pi}{20} + i\sin\frac{\pi}{20} \right)^{5}} =\]

\[= \left( \cos\frac{\pi}{20} + i\sin\frac{\pi}{20} \right)^{- 5} =\]

\[= \cos\left( - 5 \bullet \frac{\pi}{20} \right) + i\sin\left( - 5 \bullet \frac{\pi}{20} \right) =\]

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

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