How do you find the Cartesian equation from these parametric equations?
$$x=\cos2\theta \quad\text{and}\quad y=1+\sin2\theta$$
I have eliminated $\theta$ by rearranging $x$ to get $\theta=\frac{1}{2}\cos^{-1}(x)$ and then substituting this into $y$ to get $$y=1+\sin(\cos^{-1}(x)) \tag1$$
However the relationship between $x$ and $y$ is given as $$x^2+y^2-2y=0 \tag{2}$$
How do you get from $(1)$ to $(2)$?
Best Answer
If $x=\cos(2\theta)$ and $y=1+\sin(2\theta)$, then$$x^2+(y-1)^2=1,$$which is the same thing as$$x^2+y^2-2y=0.$$