# 8. Curves in Polar Coordinates

We'll plot the graphs in this section using a computer. You'll also learn how to sketch some of them on paper because it helps you understand how graphs in polar coordinates work.

Don't worry about all the difficult-looking algebra in the second part of the answers - it's just there to demonstrate that polar coordinates are much simpler than rectangular coordinates for these graphs. We convert them using what we learned in the last section, Polar Coordinates.

## Helpful Background

Curves in polar coordinates work very similarly to vectors. See:

Vector concepts

### Need Graph Paper?

(Polar graph paper included.)

Sketch each of the following functions using polar coordinates, and then convert each to an equation in rectangular coordinates.

(1) r = 2 + 3 sin θ

(This polar graph is called a limacon from the Latin word for "snail".)

Here's another example of a limacon:

(2) r = 3 cos 2θ

(3) r = sin θ − 1

(This one is called a cardioid because it is heart-shaped. It is a special case of the limacon.)

Continues below

(4) r = 2.5

(5) r = 0.2 θ

This is an interesting curve, called an Archimedean Spiral. As θ increases, so does r.

See also Equiangular Spiral.

Later, we'll learn how to find the Length of an Archimedean Spiral.

(6) r = sin (2θ) − 1.7

This is the face I drew at the top of this page. We're not even going to try to find the equivalent in rectangular coordinates!

### Application

Check out Polar Coordinates and Cardioid Microphones for an application of polar coordinates.

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