From parametric to symmetric form

Question

I'm studying parametric equations and the manual I'm following does not explain how it goes from parametric to symmetric form. The problem is:

$$ x(t)=3\cos^2 t\\ y(t)=3\sin^2 t $$

I only know that the solution is $x+y=3$. Until now, to deal with this kind of problem, I just replace the dummy variable on the other equation and I got the symmetric equation. But now, I have no idea how to deal with the cos and sin squares. So, how I go from parametric to symmetric equations in this case?

Thanks!

Answer 1

Note that $$ x(t) + y(t) = 3 \cos^2 t + 3 \sin^2 t = 3 \left(\cos^2 t + \sin^2 t\right) = 3 \cdot 1 = 3, $$ so the resulting curve is $x+y=3$.

Note that as $t$ travels from $0$ to $2\pi$, you have $0 \le x,y \le 3$, so the resulting curve is a line segment, even if you consider larger ranges of $t$.