plot the point $(-5,2\\pi)$, given in polar coordinates, and find other polar coordinates $(r,\\theta)$ of…

Explanation:

Step1: Recall the rules for polar coordinates

For a polar coordinate ((r,\theta)), if we change (r) to (-r), we add (\pi) to (\theta) (or subtract (\pi)). Also, ((r,\theta)=(r,\theta + 2k\pi)) for any integer (k).

Step2: Solve part (a)

Given ((- 5,2\pi)). We want (r>0), so (r = 5). And (-2\pi\leq\theta<0). Since ((-r,\theta)=(r,\theta-\pi)) (when changing the sign of (r)). Here (\theta = 2\pi-\pi-\pi=0-\pi). (\theta=2\pi - 3\pi=-\pi). So the coordinate is ((5,-\pi))

Step3: Solve part (b)

We want (r < 0), (0\leq\theta<2\pi). Starting from ((-5,2\pi)), if we use the formula ((r,\theta)=(r,\theta + 2k\pi)) with (k = 0) and keep (r=-5), (\theta = 2\pi) is not in the range (0\leq\theta<2\pi). Using ((-r,\theta)=(r,\theta-\pi)) (equivalent transformation for polar coordinates). If (r=-5), and we know ((-5,2\pi)=(5,2\pi-\pi)=(5,\pi)), then ((-5,2\pi)=(-5,0)) (because ((r,\theta)=(r,\theta + 2k\pi)), when (k=-1) for ((5,\pi)) ((5,\pi)=( - 5,\pi+\pi)))

Step4: Solve part (c)

We want (r>0), (2\pi\leq\theta<4\pi) Starting from ((-5,2\pi)), since ((-r,\theta)=(r,\theta-\pi)), and then using ((r,\theta)=(r,\theta + 2k\pi)) ((-5,2\pi)=(5,2\pi-\pi)=(5,\pi)), then ((5,\pi)=(5,\pi + 2\pi)=(5,3\pi))

Answer:

(a) ((5,-\pi)) (b) ((-5,0)) (c) ((5,3\pi))