Convert $x(t)$ and $y(t)$ into Cartesian form

Question

Consider the parametric equations given by \begin{align*} x(t)&=\sin{t}-t,\\ y(t) & = 1-\cos{t}.\end{align*}

I want to write these parametric equations in Cartesian form.

In order to eliminate the sine and cosine terms I think I probably need to consider some combination of $x(t),y(t), x(t)^2$ and $y(t)^2$ but I can't see exactly how to do this.

Answer 1

From the second equality, you'll get $t=\arccos(1-y)$. So,\begin{align}x&=\sin\bigl(\arccos(1-y)\bigr)-\arccos(1-y)\\&=\sqrt{1-(1-y)^2}-\arccos(1-y)\\&=\sqrt{2y-y^2}-\arccos(1-y).\end{align}

Answer 2

$$t=\arccos(1-y)$$ $$x=\sin(\arccos(1-y))-\arccos(1-y)$$ $$=\sqrt{1-(1-y)^2}-\arccos(1-y)$$

Answer 3

$$ y(t) = 1-\cos{t}\implies t= \arccos(1-y) \implies x= \arccos(\sin(1-y)) -\arccos(1-y)$$ that is

$$x= \sqrt{1-(1-y)^2} -\arccos(1-y)$$