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,

So the point ​z​ has coordinates

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,

So the point ​z​ has coordinates