Question: Plot the point \((-6, 3\pi)\), given in polar coordinates, and find other polar coordinates…

Plot the point \((-6, 3\pi)\), given in polar coordinates, and find other polar coordinates \((r, \theta)\) of this same point for which the following are true.

(a) \(r > 0\), \(-2\pi \leq \theta < 0\)

(b) \(r < 0\), \(0 \leq \theta < 2\pi\)

(c) \(r > 0\), \(2\pi \leq \theta < 4\pi\)

Solution

Let’s solve the problem by finding equivalent polar coordinates for the point \((-6, 3\pi)\). ### (a) \( r > 0, -2\pi \leq \theta < 0 \) The given point is \((-6, 3\pi)\). To make \(r > 0\), we change the sign of \(r\) and add \(\pi\) to \(\theta\): \[ r = 6, \quad \theta = 3\pi + \pi = 4\pi \] Since we need \(-2\pi \leq \theta < 0\), subtract \(2\pi\) repeatedly: \[ \theta = 4\pi - 2\pi = 2\pi \] \[ \theta = 2\pi - 2\pi = 0 \] Thus, the coordinates are: \[ (r, \theta) = (6, 0) \] ### (b) \( r < 0, 0 \leq \theta < 2\pi \) Given that \(r\) is already negative, we can leave \(r\) as \(-6\). Adjust \(\theta\) to be within the desired range: Subtract \(2\pi\): \[ \theta = 3\pi - 2\pi = \pi \] Thus, the coordinates are: \[ (r, \theta) = (-6, \pi) \] ### (c) \( r > 0, 2\pi \leq \theta < 4\pi \) For \(r > 0\), again change the sign of \(r\) and add \(\pi\) to \(\theta\): \[ r = 6, \quad \theta = 3\pi + \pi = 4\pi \] This \(\theta\) is already within the required range. So the coordinates are: \[ (r, \theta) = (6, 4\pi) \] These are the equivalent coordinates for each case.