Given the point \(\left(4, \frac{7 \pi}{6}\right)\) in polar coordinates, we need to find other polar coordinates \((r, \theta)\) for the point under three different conditions:
### (a) \(r > 0, -2\pi \leq \theta < 0\)
First, we need to transform \(\theta = \frac{7\pi}{6}\) to fit within the range \(-2\pi \leq \theta < 0\).
By subtracting \(2\pi\) from \(\theta\),
[tex]\[
\theta_a = \frac{7 \pi}{6} - 2\pi = \frac{7\pi}{6} - \frac{12\pi}{6} = \frac{-5\pi}{6}
\][/tex]
So, the coordinates are:
[tex]\[
(4, -\frac{5\pi}{6})
\][/tex]
### (b) \(r < 0, 0 \leq \theta < 2\pi\)
For a negative radius, \(r\), we need to add \(\pi\) to \(\theta\) to adjust the angle appropriately.
[tex]\[
r_b = -4
\][/tex]
[tex]\[
\theta_b = \frac{7\pi}{6} + \pi = \frac{7\pi}{6} + \frac{6\pi}{6} = \frac{13\pi}{6}
\][/tex]
Since \(\frac{13\pi}{6}\) is within \(0\) to \(2\pi\) (as \(\frac{13\pi}{6} \approx 6.8068\)), the coordinates are:
[tex]\[
(-4, \frac{13\pi}{6})
\][/tex]
### (c) \(r > 0, 2\pi \leq \theta < 4\pi\)
For a positive radius \(r\) where \(\theta\) is within the range \(2\pi \leq \theta < 4\pi\), we add \(2\pi\) to \(\theta\).
[tex]\[
\theta_c = \frac{7\pi}{6} + 2\pi = \frac{7\pi}{6} + \frac{12\pi}{6} = \frac{19\pi}{6}
\][/tex]
Since \(\frac{19\pi}{6}\) is within the required range (as \(\frac{19\pi}{6} \approx 9.9484\)), the coordinates are:
[tex]\[
(4, \frac{19\pi}{6})
\][/tex]
To summarize:
(a) For \(r > 0, -2\pi \leq \theta < 0\): \((4, -\frac{5\pi}{6})\)
(b) For \(r < 0, 0 \leq \theta < 2\pi\): \((-4, \frac{13\pi}{6})\)
(c) For [tex]\(r > 0, 2\pi \leq \theta < 4\pi\)[/tex]: [tex]\((4, \frac{19\pi}{6})\)[/tex]
