## Task A-3.3. (8 points)

Given a complex number $z=1+\cos \alpha+i \sin \alpha$, where $\alpha \in \mathbb{R}$.

Determine $|z|$. Write the solution without the square root symbol.
## Solution.

$|z|=\sqrt{(1+\cos \alpha)^{2}+(\sin \alpha)^{2}}=\sqrt{2+2 \cos \alpha}=2\left|\cos \frac{\alpha}{2}\right|$.

For 4 points: final result without the absolute value: $2 \cos \frac{\alpha}{2}$.

For 8 points: correct result and procedure.