Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 633

\[1)\ \frac{\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}} =\]

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

\[= \cos\frac{\pi}{6} + i\sin\frac{\pi}{6};\]

\[2)\ \frac{8\left( \cos\frac{5\pi}{4} + i\sin\frac{5\pi}{4} \right)}{2\left( \cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4} \right)} =\]

\[= \frac{8}{2}\left( \cos\left( \frac{5\pi}{4} - \frac{3\pi}{4} \right) + i\sin\left( \frac{5\pi}{4} - \frac{3\pi}{4} \right) \right) =\]

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

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

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

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

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

\[4)\ \frac{\cos{30{^\circ}} + i\sin{30{^\circ}}}{2(\cos( - 15{^\circ}) + i\sin( - 15{^\circ})} =\]

\[= \frac{1}{2}\left( \cos(30{^\circ} + 15{^\circ}) + i\sin(30{^\circ} + 15{^\circ}) \right) =\]

\[= \frac{1}{2}\left( \cos{45{^\circ}} + i\sin{45{^\circ}} \right);\]

\[5)\ \frac{\sqrt{12}\left( \cos{20{^\circ}} + i\sin{20{^\circ}} \right)}{\sqrt{3}\left( \cos{50{^\circ}} + i\sin{50{^\circ}} \right)} =\]

\[= \frac{\sqrt{12}}{\sqrt{3}}\left( \cos(20{^\circ} - 50{^\circ}) + i\sin(20{^\circ} - 50{^\circ}) \right) =\]

\[= 2\left( \cos( - 30{^\circ}) + i\sin( - 30{^\circ}) \right);\]

\[6)\ \frac{\cos 7 + i\sin 7}{\cos 2 + i\sin 2} =\]

\[= \cos(7 - 2) + i\sin(7 - 2) =\]

\[= \cos 5 + i\sin 5.\]