Complex analysis clarification Let us look upon a complex number "z".
$$
z=x+iy
$$
The following are true:
$$
\bar z = x-iy, \quad |z|=\sqrt{x^2+y^2}
$$
Since $z$ is a complex number it is represented in the complex plane (I hope this is the correct nomenclature) by the vector $\vec{z}$, which can be defined by its length and the angle it concludes(?) with the positive axis $O_x$ . The angle is called an "argument" of $z$ and is denoted as $\arg{z}$ . Clearly if $\phi$ is an argument of $z$ then $\phi+2k\pi, k\in \Bbb Z$ is also an argument of $z$. Amongst all arguments of $z$ there is but one, which belongs to the $(-\pi,\pi]$ range, it is denoted by $\arg_0$ and is called "Main argument" of $z$.

First question here: is it not the range $(-\pi,\pi]$ the entirety of  all angles, past and present? What is the point of saying there is but one angle in the entire realm of angles that is the "main" argument of $z$ if any argument of $z$ to which 360° are added is also an argument of $z$, is it just to mark the territory of the Primal angle or something?

Moving on...
As we said - $z=x+iy$ , and if $r = |z|$ and $\phi = \arg{z}$ then $x = r\cos{\phi}$, $y = r\sin{\phi}$ and $z = r(\cos{\phi} + i\sin{\phi})$ and this we shall call the "trigonometric form" of the number. For convenience we shall introduce Euler's notation $e^{i\phi} = \cos{\phi} + i\sin{\phi}$, therefore $z = re^{i\phi}$
Now if we were to express the number $1+\cos{\alpha}+i\sin{\alpha}$ by Euler's notation we would get (after some rudimentary trigonometric transformations) $2\cos{\frac{\alpha}{2}} e^{i \alpha/2}$. Now if $\alpha \in [0, \pi]$ all is hunky-dory (because $\cos{\frac{\alpha}{2}} \ge 0$), however if $\alpha \in [\pi , 2\pi]$ then according to my textbook $\cos{\frac{\alpha}{2}} \le 0$ and then $z = -2\lvert \cos{\frac{\alpha}{2}}\rvert e^{i\alpha/2}$ and $z = 2\lvert \cos{\frac{\alpha}{2}}\rvert e^{i(\pi+\alpha/2)}$.

Second question: why would we need to put a minus sign in front of the "2" AND a modulus on the $\cos{\frac{\alpha}{2}}$? Is it because we expect the $e^{i \alpha/2}$ to be negative, as well, even though it is a sum of a $\cos$ and a $\sin$ and only one of those would be negative? Secondly I do not understand what is the problem with having a negative complex number, or is it that since we are attempting to transform it into trigonometrical form, we are trying to transform it to a trigonometrical number and those cannot be negative? 

 A: The principal or main argument of the complex number is used such that $\arg_0(z)$ is a well-defined function. It can avoid ambiguity. For example, the number $i$ has argument $2k\pi+\frac{\pi}{2}, k\in \mathbb{Z}$. Its main argument is $\frac{\pi}{2}$. It assigns only one value to each complex number, so it's well-defined. 
The polar form, or trig form, of a complex number should be $re^{i\theta}$, where $r>0$. So when the number before $e^{}$ is negative, you need to transform it to positive, such that it represents the "magnitude" of the complex number. At the same time, when you flip the sign of the magnitude, the direction of the complex number is reversed, so you add a $\pi$ to the argument.
In your example, the $z = -2|cos\frac\alpha2|e^{i\frac\alpha2}$ is an intermediate step for you to see that the magnitude is negative and needs to be transformed to a positive value. 
Here is another example. Consider the complex number $-1+i$, its polar form is $\sqrt{2}e^{i\frac{3}{4}\pi}$, of course it can also be represented by $-\sqrt{2}e^{-i\frac{\pi}{4}}$, $\sqrt{2}e^{-i\frac{5}{4}\pi}$, $-\sqrt{2}e^{i\frac{7}{4}\pi}$, etc. But only $\sqrt{2}e^{i\frac{3}{4}\pi}$ is called the POLAR FORM with the main argument of it by DEFINITION. In mathematics, definition is very important. Definitions are used to avoid confusion, to avoid ambiguity, to prove theorems, etc. Also from this example you can see where the definition of main argument is important. It well-defines the polar form. 
See the following picture for this example:

If you treat it as a vector, when the magnitude is $2$, it points to upper left direction. If you say the "magnitude" is $-2$, the direction would be reversed. 
However the polar form of a complex number is not really a vector. It is a concept that defines magnitude to be positive and angle to be the main argument. 
I hope this helps.
