How do you parameterize a circle?

Question

I need some help understanding how to parameterize a circle.

Suppose the line integral problem requires you to parameterize the circle, $x^2+y^2=1$. My question is, if I parameterize it, would it just be: $r(t)=($cos $t)i+($sin $t)j$? And how would that change if say my radius became $4$ instead of $1$? Thanks in advance!

Answer 1

If you have the circle $x^2+y^2=a^2$, then $x=a$cos $t$ and $y=a$ sin $t$ where $0\leq t \leq 2\pi$. Then $r(t)=(a$ cos $t)i+(a$ sin $t)j$.

Answer 2

Your parametrization is correct. Once you have a parameterization of the unit circle, it's pretty easy to parameterize any circle (or ellipse for that matter): What's a circle of radius $4$? Well, it's four times bigger than a circle of radius $1$!