# 8. Curves in Polar Coordinates

*r* = sin (2^{θ}) − 1.7

This is a real graph using polar coordinates.

Okay, I admit to adding the
eye and smile. :-)

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:

### Examples

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 2*θ*

(3) *r* = sin *θ* − 1

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

(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 (2^{θ}) − 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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