Any point z in the plane sits at the end of a segment from the origin of length ∣z∣, making an angle arg(z) with the positive real axis. If you drop a perpendicular from z down to the real axis, you form a right-angled triangle whose – horizontal side is the “run” x, – vertical side is the “rise” y, – hypotenuse is the distance ∣z∣.
By basic trigonometry in that triangle,
x=∣z∣cos(arg(z)),y=∣z∣sin(arg(z)).
So the point z has coordinates
(∣z∣cos(arg(z)),∣z∣sin(arg(z))).