10.49 Find the modulus of the complex number $z^{3}+z^{5}$, if $z=\cos \alpha+i \sin \alpha, \alpha \in\left(\pi, \frac{3 \pi}{2}\right)$.
10.49 We have

$$
\begin{aligned}
& \left|z^{3}+z^{5}\right|=|\cos 3 \alpha+i \sin 3 \alpha+\cos 5 \alpha+i \sin 5 \alpha|= \\
& =|\cos 3 \alpha+\cos 5 \alpha+i(\sin 3 \alpha+\sin 5 \alpha)|= \\
& =\sqrt{4 \cos ^{2} 4 \alpha \cos ^{2} \alpha+4 \sin ^{2} 4 \alpha \cos ^{2} \alpha}= \\
& =\sqrt{4 \cos ^{2} \alpha}=2|\cos \alpha|, \text { since } \alpha \in\left(\pi, \frac{3 \pi}{2}\right) .
\end{aligned}
$$

Answer: $2|\cos \alpha|$.