Question

suppose a point has polar coordinates \\(\left(6, -\frac{\pi}{2}\
ight)\\), with the angle measured in radians. find two additional polar representations of the point. write each coordinate in simplest form with the angle in \\(-2\pi, 2\pi\\).

Explanation:

Step1: Add \(2\pi\) to the angle

For a polar coordinate \((r,\theta)\), \((r,\theta + 2k\pi)\) (\(k\in\mathbb{Z}\)) represents the same point.
Let \(r = 6\) and \(\theta=-\frac{\pi}{2}\). When \(k = 1\), \(\theta+2\pi=-\frac{\pi}{2}+2\pi=\frac{3\pi}{2}\). So one polar representation is \((6,\frac{3\pi}{2})\).

Step2: Use \((-r,\theta+(2k + 1)\pi)\)

Another form of representing the same polar point is \((-r,\theta+(2k + 1)\pi)\) (\(k\in\mathbb{Z}\)).
Let \(r = 6\), \(\theta=-\frac{\pi}{2}\) and \(k = 0\). Then \(-r=-6\) and \(\theta+\pi=-\frac{\pi}{2}+\pi=\frac{\pi}{2}\). So another polar representation is \((-6,\frac{\pi}{2})\).

Answer:

\((6,\frac{3\pi}{2})\), \((-6,\frac{\pi}{2})\)