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

To find other polar coordinates \((r, \theta)\) for the point \((-6, 3\pi)\), we need to adjust \(r\) and \(\theta\) to fit the given conditions. ### (a) \(r > 0\), \(-2\pi \leq \theta < 0\) The original point is \((-6, 3\pi)\). Change the signs: \(r = 6\) To maintain the same point, add \(\pi\) (180 degrees) to the angle because the radius was negative: \[ 3\pi - \pi = 2\pi \] Subtract \(2\pi\) to bring the angle within the desired range: \[ 2\pi - 2\pi = 0 \] Now, subtract another \(2\pi\): \[ 0 - 2\pi = -2\pi \] Coordinates: \( (6, -2\pi) \) ### (b) \(r < 0\), \(0 \leq \theta < 2\pi\) Start from \((-6, 3\pi)\). The radius is already negative: \(r = -6\) Subtract \(2\pi\) from the angle to fit within the range: \[ 3\pi - 2\pi = \pi \] Coordinates: \( (-6, \pi) \) ### (c) \(r > 0\), \(2\pi \leq \theta < 4\pi\) Start from \((-6, 3\pi)\). Change the sign for a positive radius: \(r = 6\) Add \(\pi\) to convert the negative radius: \[ 3\pi - \pi = 2\pi \] Coordinates: \( (6, 2\pi) \)