Converting a Parametric equation into a Cartesian one

Question

I was working on converting an parametric equation into a Cartesian one and i cant seem to figure this one out. I was hoping you could help with that for this equation of a cycloid, Thanks

$x = cos(t)+t+\pi$

$y = cos(t)$

Edit: Oops I gave you the wrong equation

$x = sin(t)+t+\pi$

$y = cos(t)$

Sorry About that, And Thanks for the many Answers

Answer 1

If you want to eliminate $t$ you have$$x-y=t+\pi\Rightarrow \cos(x-y)=\cos(t+\pi)=-\cos t=-y$$

Hence $$\cos(x-y)+y=0$$

Answer 2

By subtracting one equation from other to eliminate $ \cos t$,

$$ y-x = t + \pi$$

Now $t$ has to be also eliminated

$$ y-x-\pi = \cos^{-1}y $$

Take cos and simplify, getting

$$ \cos ( x-y) + y =0 $$

which is the required equation in rectangular coordinates ... an implicit equation.

Answer 3

If the equations are $$x = \sin(t)+t+\pi\qquad ,\qquad y=\cos(t)$$ you have $$\sin(t)=x-t-\pi$$ $$\cos(t)=y$$ $$\sin^2(t)+\cos^2(t)=(x-t-\pi)^2+y^2=1$$ but $t=\cos^{-1}(y)$. So, in cartesian coordinates $$(x-\cos^{-1}(y)-\pi)^2+y^2=1$$ is the implicit equation.