Answer
$x = 2t-2~sin~t$
$y = 2-2~cos~t$
$0 \leq t \leq 4\pi$
We can see the graph below.
Work Step by Step
$x = 2t-2~sin~t$
$y = 2-2~cos~t$
$0 \leq t \leq 4\pi$
When $t = 0$:
$x = 2(0)-2~sin~0 = 0$
$y = 2-2~cos~0 = 0$
When $t = \frac{\pi}{6}$:
$x = 2(\frac{\pi}{6})-2~sin~\frac{\pi}{6} = 0.047$
$y = 2-2~cos~\frac{\pi}{6} = 0.268$
When $t = \frac{\pi}{4}$:
$x = 2(\frac{\pi}{4})-2~sin~\frac{\pi}{4} = 0.157$
$y = 2-2~cos~\frac{\pi}{4} = 0.586$
When $t = \frac{\pi}{3}$:
$x = 2(\frac{\pi}{3})-2~sin~\frac{\pi}{3} = 0.362$
$y = 2-2~cos~\frac{\pi}{3} = 1$
When $t = \frac{\pi}{2}$:
$x = 2(\frac{\pi}{2})-2~sin~\frac{\pi}{2} = 1.14$
$y = 2-2~cos~\frac{\pi}{2} = 2$
When $t = \frac{2\pi}{3}$:
$x = 2(\frac{2\pi}{3})-2~sin~\frac{2\pi}{3} = 2.46$
$y = 2-2~cos~\frac{2\pi}{3} = 3$
When $t = \pi$:
$x = 2(\pi)-2~sin~\pi = 6.28$
$y = 2-2~cos~\pi = 4$
When $t = \frac{4\pi}{3}$:
$x = 2(\frac{4\pi}{3})-2~sin~\frac{4\pi}{3} = 10.110$
$y = 2-2~cos~\frac{4\pi}{3} = 3$
When $t = \frac{3\pi}{2}$:
$x = 2(\frac{3\pi}{2})-2~sin~\frac{3\pi}{2} = 11.425$
$y = 2-2~cos~\frac{3\pi}{2} = 2$
When $t = 2\pi$:
$x = 2(2\pi)-2~sin~2\pi = 12.566$
$y = 2-2~cos~2\pi = 0$
When $t = 3\pi$:
$x = 2(3\pi)-2~sin~3\pi = 18.850$
$y = 2-2~cos~3\pi = 4$
When $t = 4\pi$:
$x = 2(4\pi)-2~sin~4\pi = 25.133$
$y = 2-2~cos~4\pi = 0$
We can see the graph below.
