Parameter & Cartesian Equation

Question

How would I eliminate the parameter and solve the equation of the curve (y=...) within an x-interval?

Answer 1

I believe you can solve this way:

Divide x=2sin(theta) by 2 on both sides -----> sin(theta)=x/2. Rewrite y=cos(theta)^2 to be 1 - sin(theta)^2. Then substitute x/2 in for sin(theta) of the y equation to now read y = 1 - (x/2)^2 = 1 - (1/4)x^2. Substitute the values of the theta interval into the original x equation to get x = 2sin(-pi/2) = -2 & x = 2sin(pi/2) = 2.

So the x-interval is -2 <= x <= 2, and the y equation (as a function of x) = 1 - (1/4)x^2.