Answer
$x = 3~cos~2t$
$y = 3~sin~3t$
$0 \leq t \leq 6.5$
We can see the graph in the window $[-6,6]$ by $[-4,4]$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/93da5364-3bac-4cf6-be06-4ae761c66062/result_image/1555898542.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240908%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240908T000534Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=fa90fc408b3e624c0a3490c43c9aef7ff66cca78e88bca6bf756ecb617d609d4)
Work Step by Step
$x = 3~cos~2t$
$y = 3~sin~3t$
$0 \leq t \leq 6.5$
When $t = 0$:
$x = 3~cos~0 = 3$
$y = 3~sin~0 = 0$
When $t = \frac{\pi}{12}$:
$x = 3~cos~\frac{2\pi}{12} = 2.6$
$y = 3~sin~\frac{3\pi}{12} = 2.12$
When $t = \frac{\pi}{6}$:
$x = 3~cos~\frac{2\pi}{6} = 1.5$
$y = 3~sin~\frac{3\pi}{6} = 3$
When $t = \frac{\pi}{4}$:
$x = 3~cos~\frac{2\pi}{4} = 0$
$y = 3~sin~\frac{3\pi}{4} = 2.12$
When $t = \frac{\pi}{3}$:
$x = 3~cos~\frac{2\pi}{3} = -1.5$
$y = 3~sin~\frac{3\pi}{3} = 0$
When $t = \frac{\pi}{2}$:
$x = 3~cos~\frac{2\pi}{2} = -3$
$y = 3~sin~\frac{3\pi}{2} = -3$
When $t = \frac{2\pi}{3}$:
$x = 3~cos~\frac{4\pi}{3} = -1.5$
$y = 3~sin~\frac{6\pi}{3} = 0$
When $t = \frac{3\pi}{4}$:
$x = 3~cos~\frac{6\pi}{4} = 0$
$y = 3~sin~\frac{9\pi}{4} = 2.12$
When $t = \pi$:
$x = 3~cos~2\pi = 3$
$y = 3~sin~3\pi = 0$
When $t = \frac{7\pi}{6}$:
$x = 3~cos~\frac{14\pi}{6} = 1.5$
$y = 3~sin~\frac{21\pi}{6} = -3$
When $t = \frac{5\pi}{4}$:
$x = 3~cos~\frac{10\pi}{4} = 0$
$y = 3~sin~\frac{15\pi}{4} = -2.12$
When $t = \frac{7\pi}{3}$:
$x = 3~cos~\frac{14\pi}{3} = -1.5$
$y = 3~sin~\frac{21\pi}{3} = 0$
When $t = \frac{3\pi}{2}$:
$x = 3~cos~\frac{6\pi}{2} = -3$
$y = 3~sin~\frac{9\pi}{2} = 3$
When $t = \frac{7\pi}{4}$:
$x = 3~cos~\frac{14\pi}{4} = 0$
$y = 3~sin~\frac{21\pi}{4} = -2.12$
When $t = 2\pi$:
$x = 3~cos~4\pi = 3$
$y = 3~sin~6\pi = 0$
When $t = 6.5$:
$x = 3~cos~13 = 2.72$
$y = 3~sin~19.5 = 1.82$
We can see the graph in the window $[-6,6]$ by $[-4,4]$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/93da5364-3bac-4cf6-be06-4ae761c66062/steps_image/small_1555898542.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240908%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240908T000534Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=bd6e1cf3222e8dcfae61e2f2ca66049f10b8cd417cd37197fcacc7302c6fb158)