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Polar Equation of a Circle

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Date: 04/12/97 at 16:26:52
From: Frank P.
Subject: Polar Equation of a Circle

I was wondering if you could provide for me a description of the
following polar equations for a circle.

r = a*sin (theta)  and r = a*cos (theta).

What influence does a have on the graph?  What influence does theta
have on the graph?  What's the difference in using sin or cos, and
how do they change the graph?
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Date: 04/12/97 at 17:34:07
From: Doctor Mitteldorf
Subject: Re: Polar Equation of a Circle

Dear Frank,

You are familiar with writing an equation in x and y and then
plotting the curve that represents all points obeying that equation.
Well, you can do the same thing using r and theta instead of x and y.

In fact, r and theta are related to x and y by

x = r*cos(theta) and y = r*sin(theta).

Equivalently,

r^2 = x^2 + y^2 and tan(theta) = y/x.

Another way to write this second equation is

sin(theta) = y/sqrt(x^2+y^2).

The obvious way to write the equation for a circle in polar
coordinates is r = a.  Then theta can be anything.  This makes a
circle of radius a around the origin as center.

The equation you've written,  r = a sin(theta), is also a circle, but
it's not so obvious, and it's not centered about the origin.  It's
centered around another point.  In a few minutes, you should be
prepared to answer what that point is, and what the radius of the
circle is.

>what influence does a have on the graph?

We should be able to answer this in a minute

>what influence does theta have on the graph?

This isn't a meaningful question.  Theta is one of the variables in
the equation, like x and y were in the cartesian equation.

To see that you have a circle, and to see where the center and the
radius are, just substitute into your equation r=a sin(theta).

Let

r = sqrt(x^2+y^2)               and let
sin(theta) = y/sqrt(x^2+y^2).

Rearrange the equation and you'll have the form

x^2 + (y-a/2)^2 = (a/2)^2.

When you see the equation in that form, it should be clear what the
radius is and where the center is.

Let me know if this is not clear, please.

-Doctor Mitteldorf,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Equations, Graphs, Translations

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