Answer
$x = 2~cos~t$
$y = 3~sin~2t$
$0 \leq t \leq 6.5$
We can see the graph in the window $[-6,6]$ by $[-4,4]$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/7a83f22c-6aff-4638-9107-7d8b02e66970/result_image/1555865063.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240908%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240908T000244Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=6da9c0a26234c720c0c3a02a8d202eab0d1d2477202daa0bd1b39ec5dc05e068)
Work Step by Step
$x = 2~cos~t$
$y = 3~sin~2t$
$0 \leq t \leq 6.5$
When $t = 0$:
$x = 2~cos~0 = 2$
$y = 3~sin~[2(0)] = 0$
When $t = \frac{\pi}{4}$:
$x = 2~cos~\frac{\pi}{4} = 1.41$
$y = 3~sin~[2(\frac{\pi}{4})] = 3$
When $t = \frac{\pi}{3}$:
$x = 2~cos~\frac{\pi}{3} = 1$
$y = 3~sin~[2(\frac{\pi}{3})] = 2.6$
When $t = \frac{\pi}{2}$:
$x = 2~cos~\frac{\pi}{2} = 0$
$y = 3~sin~[2(\frac{\pi}{2})] = 0$
When $t = \frac{2\pi}{3}$:
$x = 2~cos~\frac{2\pi}{3} = -1$
$y = 3~sin~[2(\frac{2\pi}{3})] = -2.6$
When $t = \frac{3\pi}{4}$:
$x = 2~cos~\frac{3\pi}{4} = -1.41$
$y = 3~sin~[2(\frac{3\pi}{4})] = -3$
When $t = \pi$:
$x = 2~cos~\pi = 2$
$y = 3~sin~(2\pi) = 0$
When $t = \frac{5\pi}{4}$:
$x = 2~cos~\frac{5\pi}{4} = -1.41$
$y = 3~sin~[2(\frac{5\pi}{4})] = 3$
When $t = \frac{3\pi}{2}$:
$x = 2~cos~\frac{3\pi}{2} = 0$
$y = 3~sin~[2(\frac{3\pi}{2})] = 0$
When $t = \frac{7\pi}{4}$:
$x = 2~cos~\frac{7\pi}{4} = 1.41$
$y = 3~sin~[2(\frac{7\pi}{4})] = -3$
When $t = 2\pi$:
$x = 2~cos~2\pi = 2$
$y = 3~sin~(4\pi) = 0$
When $t = 6.5$:
$x = 2~cos~6.5 = 1.95$
$y = 3~sin~13 = 1.26$
We can see the graph in the window $[-6,6]$ by $[-4,4]$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/7a83f22c-6aff-4638-9107-7d8b02e66970/steps_image/small_1555865063.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240908%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240908T000244Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=629e1c0bee4ad67f8cf4e812e5a277c5eeba1702039a2049e2688c7fe405843e)