How do you find the parametric equations of a circle?

1 Answer
Aug 23, 2014

We'll start with the parametric equations for a circle:

y = rsin t
x = rcos t

where t is the parameter and r is the radius.

If you know that the implicit equation for a circle in Cartesian coordinates is x^2 + y^2 = r^2 then with a little substitution you can prove that the parametric equations above are exactly the same thing.

We will take the equation for x, and solve for t in terms of x:

x/r = cos t
t = arccos (x/r)

Now substitute into the equation for y. This eliminates the parameter t and gives us an equation with only x and y.

y = rsin arccos(x/r)

sin arccos(x/r) is equal to sqrt(r^2 - x^2)/r. This is apparent if one sketches a right triangle, letting theta = arccos(x/r):

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Thus, sin theta = sqrt(r^2 - x^2)/r. So now we have

y = r*sqrt(r^2 - x^2)/r

This simplifies to

y = sqrt(r^2 - x^2)

If we square this entire deal and solve for r, we get:

r^2 = x^2 + y^2

which is precisely the equation for a circle in Cartesian coordinates.