I came across a parametric equation: $$x=2t-2\sin (t)\\y=2-2\cos(t)$$ The question asked me to graph it on a calculator, so I graphed it. But then I questioned: is there a way to write it explicitly (like $y=f(x)$)? I tried but couldn't get the $2t$ and the trig function away the same time. In general, can I parametric equation be written explicitly?
Any help appreciated.
Best Answer
First write $x-2t=-2\sin t,\,y-2=-2\cos t.$ Then square and add to get $$(x-2t)^2+(y-2)^2=4.$$ Finally, solve for $t$ in the second equation to get $\cos t=\frac{2-y}{2}.$ Then for values of $t$ such that $0\le t\le π,$ we have $$t=\arccos\left(\frac{2-y}{2}\right),$$ so that the equation becomes $$\left(x-2\arccos\left(\frac{2-y}{2}\right)\right)^2+(y-2)^2=4.$$
Apart from the fact that this probably doesn't trace out the whole curve, since there is a restriction on $y$ so that $4\ge y\ge 0,$ it is not very pretty to look at.
We are lucky in this case to be able to get such a relationship at all. It is not always possible to do this in general.