What property of complex numbers allow us to represent it in the plane? Why can trigonometry as a geometrically defined concept be used to algebraic operations between complex numbers?  What connects the two things together and how ?
 A: There are a couple connections. First of all: you can look at $\mathbb{C}$ as being isomorphic to $\mathbb{R}^2$ with regards to addition (meaning if $z_1 = x_1+iy_1$ and $z_2=x_2+iy_2$ and we associate $z = x + iy$ with the vector $(x,y)$ then with regards to addition, these behave the same - have the "same structure" just with a different name), so it makes sense to look at complex numbers on the plane. The connection with trigonometry is not as obvious but not hard to see either and it comes from Euler's formula:
$$\exp(ix) = \cos x + i\sin x.$$
Making use of the Cartesian representation of a complex number, we can see that if $z = x+iy$, $r = \sqrt{x^2+y^2}$ and $\theta = \arctan\left(\frac{y}{x}\right)$ then we can easily make the association that $x = r\cos\theta$ and $y = r\sin\theta$. By doing this what we are doing is associating the real component of the complex number with the $x$-axis and the imaginary component with the $y$-axis in the Cartesian plane. See the figure below.

This ties into Euler's formula because we now can write our complex number as
$$z = r\cos\theta + ir\sin\theta = r\exp(i\theta).$$
Using this "polar" representation makes a lot of things very nice in the complex plane. It allows us to easily multiply complex numbers, to easily divide them and makes contour integrals fairly simple.
A: $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}...$
$e^{i\theta}=1+{i\theta}+\frac{{(i\theta)}^2}{2!}+\frac{{(i\theta)}^3}{3!}+\frac{{(i\theta)}^4}{4!}...=1+{i\theta}-\frac{{\theta}^2}{2!}-i\frac{{\theta}^3}{3!}+\frac{{\theta}^4}{4!}+i\frac{\theta^5}{5!}...=$
$=[1-\frac{{\theta}^2}{2!}+\frac{{\theta}^4}{4!}-...]+i·[\theta-\frac{{\theta}^3}{3!}+\frac{\theta^5}{5!}-...]=cos(\theta)+i·sin(\theta)$
The temptation to make circles with this is great.
When you fall in the temptation and let the Y-Axis for the imaginary part the magic comes.
It can be shown that when you multiply complex numbers the modulus multiply like in the real line, but the arguments follow the adding rule.
That fact became the algebra of complex numbers in a powerful tool to describe dilations and rotations in the plane.
$z=|z|e^{i\alpha}$
$w=|w|e^{i\beta}$
$z·w=|z||w|e^{i\alpha}e^{i\beta}=|z||w|e^{i(\alpha+\beta)}$
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VISUAL COMPLEX ANALYSIS (Tristan Needham)
PREFACE
A Parable.

Imagine a society in which the citizens are encouraged, indeed compelled up to a certain age, to read (and sometimes write) musical scores. All quite admirable. However, this society also has a very curious-few remember how it all started- and disturbing law: Music must never be listened to or performed!
Thought its importance is universally acknowledged, for some reason music is not widely appreciated in this society. To be sure, professors still excitedly pore over the great works of Bach, Wagner, end the rest, and they do their utmost to communicate to their students the beautiful meaning of what they find there, but they still become tongue-tied when brashly asked the question, "What's the point of all this?!"
In this parable, it was patently unfair and irrational to have a law forbidding would-be music students from experiencing and understanding the subject directly through "sonic intuition". But in our society of mathematicians we have such a law. It is nor written law, and those who flout it may yet prosper, but it says, Mathematics must not be visualized!
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