Absolute value and Argument of a complex number with trignometric function?

Question

I have a complex number $z = -\alpha_1 \alpha_2 \sin^{2}(\theta/2)+ i\alpha_3\sin(\theta/2)|\sin(\theta/2)|$, where $\alpha_i \in \Re, i \in 1, 2, 3.$ I wish to find the absolute value and argument of this complex number. Whether the absolute value is $\sqrt{\alpha_{1}^2\alpha_{2}^2+\alpha_{3}^2}$ and whether the argument is well-defined? For finding the argument, Should I resort to case by case analysis?

Answer 1

$z = -\alpha_1 \alpha_2 \sin^{2}(\theta/2)+ i\alpha_3\sin(\theta/2)|\sin(\theta/2)|$

For the absolute value:

$\vert z\vert^2=z\bar z=\alpha_1^2\alpha_2^2\sin^4(\theta/2)+\alpha_3^2\sin^2(\theta/2)|\sin(\theta/2)|^2$

$\vert z\vert=\sin^2(\theta/2)\sqrt{\alpha_1^2\alpha_2^2+\alpha_3^2}$

For the argument, $\arg z=\text{atan2}\left(\Im(z),\Re(z)\right)$,

we have to consider $\sin(\theta/2)/\vert\sin(\theta/2)\vert$, that it is $\pm1$ depending on the value of $\theta/2$

$\arg z= \begin{cases} \text{atan2}\left(\alpha_3,-\alpha_1\alpha_2\right)&\text{for}&4k\pi\leq\theta\lt(4k+2)\pi,\;k\in\mathbb Z\\\ \text{atan2}\left(\alpha_3,\alpha_1\alpha_2\right)&\text{for}&(4k+2)\pi\leq\theta\lt(4k+4)\pi,\;k\in\mathbb Z \end{cases}$