Given ##x^2 + (y – 2)^2 = 4 ## how do you derive a parametric equation?

The parametric equations are ##x=2costheta## and ##y=2+2sintheta##

The equation represents a circle, center ##(0,2)## and radius ##r=2##

We use the following parametric equations

##x=rcostheta##

and ##y-2=rsintheta##

Therefore,

##x^2+(y-2)^2=r^2cos^2theta+r^2sin^2theta=4##

So,

##r^2(cos^2theta+sin^2theta)=4##

##r=sqrt4=2##

As ##cos^2theta+sin^2theta=1##

The parametric equations are

##x=2costheta## and ##y=2+2sintheta##