Answer
$x = t-sin~t$
$y = 1-cos~t$
$0 \leq t \leq 4\pi$
We can see the graph below:
Work Step by Step
$x = t-sin~t$
$y = 1-cos~t$
$0 \leq t \leq 4\pi$
When $t = 0$:
$x = (0)-sin~0 = 0$
$y = 1-cos~0 = 0$
When $t = \frac{\pi}{6}$:
$x = (\frac{\pi}{6})-sin~\frac{\pi}{6} = 0.024$
$y = 1-cos~\frac{\pi}{6} = 0.134$
When $t = \frac{\pi}{4}$:
$x = (\frac{\pi}{4})-~sin~\frac{\pi}{4} = 0.078$
$y = 1-cos~\frac{\pi}{4} = 0.29$
When $t = \frac{\pi}{3}$:
$x = (\frac{\pi}{3})-sin~\frac{\pi}{3} = 0.18$
$y = 1-cos~\frac{\pi}{3} = 0.5$
When $t = \frac{\pi}{2}$:
$x = (\frac{\pi}{2})-sin~\frac{\pi}{2} = 0.57$
$y = 1-cos~\frac{\pi}{2} = 1$
When $t = \frac{2\pi}{3}$:
$x = (\frac{2\pi}{3})-sin~\frac{2\pi}{3} = 1.23$
$y = 1-cos~\frac{2\pi}{3} = 1.5$
When $t = \pi$:
$x = (\pi)-sin~\pi = 3.14$
$y = 1-cos~\pi = 2$
When $t = \frac{4\pi}{3}$:
$x = (\frac{4\pi}{3})-sin~\frac{4\pi}{3} = 5.06$
$y = 1-cos~\frac{4\pi}{3} = 1.5$
When $t = \frac{3\pi}{2}$:
$x = (\frac{3\pi}{2})-sin~\frac{3\pi}{2} = 5.71$
$y = 1-cos~\frac{3\pi}{2} = 1$
When $t = 2\pi$:
$x = (2\pi)-sin~2\pi = 6.28$
$y = 1-cos~2\pi = 0$
When $t = 3\pi$:
$x = (3\pi)-sin~3\pi = 9.42$
$y = 1-cos~3\pi = 2$
When $t = 4\pi$:
$x = (4\pi)-sin~4\pi = 12.57$
$y = 1-cos~4\pi = 0$
We can see the graph below:
