Equations using Polar Co-ordinates

Equations with polar co-ordinates, like those with Cartesian ones, are a relationship between the two co-ordinates. Graphs of these equations can be draw by drawing a line through all the points that satisfy the equation.

Note// This post assumes you can use polar co-ordinates.

Examples of some simple polar equations are

r=θ

Polar graph of r=t for 0<t<pi

and

r=3sinθ

Polar Graph of r=3sint

Tips for Sketching Polar Graphs

There are number fo ways to make it easier  to sketch polar graphs. Some of the basic methods are as follows:

1. Plot a point for every π/6
2. If the graph is a cos function it will be symmetrical about the initial line (since cos(-x) = cosx). This means you only have to calculate from 0 to π and then draw the other side using symmetry.
3. If the graph is a sin function it will be symmetrical about the line θ=π/2 so you only need to calculate from -π/2 to π/2 and then flip it.

An applet for sketching these graphs is at  http://www.ies.co.jp/math/java/calc/sg_kyok/sg_kyok.html however I recommend that you attempt to sketch the graphs for yourself first and then use it to check. Note that for some graphs it doesn’t seem to give you the sketch for the full range you enter.

Conversion to and from Cartesian Form

As with points in a 2D plane polar equations can be converted to Cartesian equations and similarly Cartesian equations can be converted to polar ones. To do this we will use the same equations as were need for the conversion of points. These are:

1. x = rcosθ
2. y=rsinθ
3. r² = x² + y²
4. θ = tan^-1(y/x)

We can then substitute the appropriate equations into the one we want to convert to find its equivlinet.

A Cartesian to polar example

consider the equation of a straight line in Cartesian co-ordinates

y=3+2x

To write this in polar form we can substitute in equations 1 and 2 to get

rsinθ = 3 + 2rcosθ

which can be re-arranged however you like.

A polar to Cartesian example

consider the polar equation

r=3sin θ

Then from 2 we know sinθ=y/r so we get

r = 3y/r

which gives

r² = 3y
but from 3 we know r² = x² + y²
so we get
x² + y² = 3y