The complex plane

The angle of a complex number is found with an inverse trigonometric function of its parts.

Finding the angles whose cosine and sine match the parts of a complex number solves a trigonometric equation.

The identity cos squared + sin squared = 1 shows the complex number cos t + i sin t has modulus 1.

The vertices of a regular polygon in a circle are complex numbers of equal modulus separated by equal angles.

The n-th roots of unity are the vertices of a regular n-gon, whose rotations are multiplications by those roots.

The point 1 + i forms a 45-45-90 triangle with the axes, so its modulus is the square root of 2 and its angle is 45 degrees.

Multiplying by i rotates a point 90 degrees, as a rotation matrix does to the coordinates.

The polar form r(cos θ + i sin θ) of a complex number uses the same distance r and angle θ as the polar coordinates of the point (a, b).

Adding force or velocity vectors is the same as adding complex numbers as points in the plane.

A complex number is a point and also a vector, and adding complex numbers adds the vectors.

The equation of a circle x squared + y squared = r squared is the set of complex numbers with modulus r.