Polar coordinates and complex numbers

So far, we have located points by how far they are across and up: ${(x,y)}$. Another way is to say how far a point is from the origin and in which direction. These polar coordinates make circles and spirals easy to describe, and they produce beautiful curves.

The same idea, applied to complex numbers, turns multiplication into rotation. It gives a quick way to find powers and roots of complex numbers.

This chapter covers the following topics:

Polar coordinates

Locating points by a distance and an angle, changing between polar and rectangular coordinates, and rewriting equations.

Graphs of polar equations

Circles, cardioids, limaçons, roses, and spirals, and how to sketch a polar graph from a table of values.

Trigonometric form of complex numbers

The complex plane, the modulus and argument of a complex number, and multiplying and dividing in trigonometric form.

De Moivre’s theorem and roots

Powers of complex numbers, and finding all the ${n}$th roots of a complex number.

Practice questions

Practice problems on every section of this chapter, with full step-by-step solutions.