Answer
$x = t+cos~t$
$y = sin~t$
$t$ in $[0,2\pi]$
We can see the graph below:
![](https://gradesaver.s3.amazonaws.com/uploads/solution/1e550ba9-0f2b-49d1-aa17-cd33343401ff/result_image/1530865879.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T014038Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=0b7f327bc4d9bdfb2f628a9dfc20dd4f8be64a4dc2eb56e16001e7a84263590d)
Work Step by Step
$x = t+cos~t$
$y = sin~t$
$t$ in $[0,2\pi]$
When $t = 0$:
$x = 0 + cos~0 = 1$
$y = sin~0 = 0$
When $t = \frac{\pi}{4}$:
$x = \frac{\pi}{4} + cos~\frac{\pi}{4} = \frac{\pi}{4} + \frac{\sqrt{2}}{2} = 1.49$
$y = sin~\frac{\pi}{4} = \frac{\sqrt{2}}{2} = 0.707$
When $t = \frac{\pi}{2}$:
$x = \frac{\pi}{2} + cos~\frac{\pi}{2} = \frac{\pi}{2}+0 = 1.57$
$y = sin~\frac{\pi}{2} = 1$
When $t = \frac{3\pi}{4}$:
$x = \frac{3\pi}{4} + cos~\frac{3\pi}{4} = \frac{3\pi}{4} -\frac{\sqrt{2}}{2} = 1.65$
$y = sin~\frac{3\pi}{4} = \frac{\sqrt{2}}{2} = 0.707$
When $t = \pi$:
$x = \pi + cos~\pi = \pi-1 = 2.14$
$y = sin~\pi = 0$
When $t = \frac{5\pi}{4}$:
$x = \frac{5\pi}{4} + cos~\frac{5\pi}{4} = \frac{5\pi}{4} -\frac{\sqrt{2}}{2} = 3.22$
$y = sin~\frac{5\pi}{4} = -\frac{\sqrt{2}}{2} = -0.707$
When $t = \frac{3\pi}{2}$:
$x = \frac{3\pi}{2} + cos~\frac{3\pi}{2} = \frac{3\pi}{2}+0 = 4.71$
$y = sin~\frac{3\pi}{2} = -1$
When $t = \frac{7\pi}{4}$:
$x = \frac{7\pi}{4} + cos~\frac{7\pi}{4} = \frac{7\pi}{4} +\frac{\sqrt{2}}{2} = 6.20$
$y = sin~\frac{7\pi}{4} = -\frac{\sqrt{2}}{2} = -0.707$
When $t = 2\pi$:
$x = 2\pi + cos~2\pi = 2\pi+1 = 7.28$
$y = sin~0 = 0$
We can see the graph below:
![](https://gradesaver.s3.amazonaws.com/uploads/solution/1e550ba9-0f2b-49d1-aa17-cd33343401ff/steps_image/small_1530865879.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T014038Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=b9531731cfe1b24c87b697d5c2f25dd1216b375806bbb035b96ccdb1ccbf8ebe)