Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 632

\[1)\ \]

\[\left( \cos\frac{\pi}{6} + i\sin\frac{\pi}{6} \right)\left( \cos\frac{\pi}{12} + i\sin\frac{\pi}{12} \right) =\]

\[= \cos\left( \frac{\pi}{6} + \frac{\pi}{12} \right) + i\sin\left( \frac{\pi}{6} + \frac{\pi}{12} \right) =\]

\[= \cos\frac{3\pi}{12} + i\sin\frac{3\pi}{12} =\]

\[= \cos\frac{\pi}{4} + i\sin\frac{\pi}{4};\]

\[2)\ \]

\[2\left( \cos\frac{\pi}{4} + i\sin\frac{\pi}{4} \right)\left( \cos\frac{\pi}{12} + i\sin\frac{\pi}{12} \right) =\]

\[= 2\left( \cos\left( \frac{\pi}{4} + \frac{\pi}{12} \right) + i\sin\left( \frac{\pi}{4} + \frac{\pi}{12} \right) \right) =\]

\[= 2\left( \cos\frac{4\pi}{12} + i\sin\frac{4\pi}{12} \right) =\]

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

\[3)\ \]

\[\sqrt{3}\left( \cos\frac{5\pi}{24} + i\sin\frac{5\pi}{24} \right) \bullet 2\left( \cos\frac{\pi}{8} + i\sin\frac{\pi}{8} \right) =\]

\[= \sqrt{3} \bullet 2\left( \cos\left( \frac{5\pi}{24} + \frac{\pi}{8} \right) + i\sin\left( \frac{5\pi}{24} + \frac{\pi}{8} \right) \right) =\]

\[= 2\sqrt{3}\left( \cos\frac{8\pi}{24} + i\sin\frac{8\pi}{24} \right) =\]

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

\[= 3 \bullet 4\left( \cos\left( \frac{\pi}{3} - \frac{\pi}{6} \right) + i\sin\left( \frac{\pi}{3} - \frac{\pi}{6} \right) \right) =\]

\[= 12\left( \cos\frac{\pi}{6} + i\sin\frac{\pi}{6} \right);\]

\[= 2\left( \cos{90{^\circ}} + i\sin{90{^\circ}} \right);\]

\[6)\ \left( \cos 7 + i\sin 7 \right)\left( \cos 3 + i\sin 3 \right) =\]

\[= \cos(7 + 3) + i\sin(7 + 3) =\]

\(= \cos 10 + i\sin 10.\)