6 - The argument(angle) of complex numbers

Remember how we said that complex numbers can be considered as vectors?
Well, for a vector, we have the magnitude(absolute value/modulus), and the angle.

In a sense, the magnitude tells a vector how long it is starting from the origin, and the angle, well, tells it where to point.

Notice how we have again, a right triangle.
So, we want to find that little angle the hypothenuse makes with the x-axis. The one that tells the vector where to point.

Let's bring back another old friend. Remember the SOHCAHTOA definition.

We want to find the angle that is opposite to b, and adjacent to a.
Well, we have an exact trigonometric function that represents the ratio of the opposite over the adjacent.
The tangent.
However, we want this angle. So let's invert it!
This angle of a complex number x, denoted by is what we call the argument of this complex number z.