Question

plot the point given in polar coordinates
$(-6,\frac{5\pi}{6})$

Explanation:

Step1: Analyze the polar coordinate components

In polar coordinates \((r,\theta)\), here \(r = - 6\) and \(\theta=\frac{5\pi}{6}\).

Step2: Understand the effect of negative \(r\)

When \(r<0\), we can find an equivalent positive - \(r\) representation. We know that \((r,\theta)=(-r,\theta +\pi)\) (or \((-r,\theta-\pi)\)). So \((-6,\frac{5\pi}{6})=(6,\frac{5\pi}{6}+\pi)=(6,\frac{11\pi}{6})\)

Step3: Locate the angle \(\frac{11\pi}{6}\)

The angle \(\theta=\frac{11\pi}{6}\) is in the fourth quadrant. It has a reference angle of \(2\pi-\frac{11\pi}{6}=\frac{\pi}{6}\).

Step4: Plot the point with \(r = 6\)

Starting from the positive \(x\) - axis, rotate an angle of \(\frac{11\pi}{6}\) (or \(330^{\circ}\)), and then move out 6 units from the origin in the direction of the terminal side of the angle.

Answer:

Plot the point by first converting \((-6,\frac{5\pi}{6})\) to \((6,\frac{11\pi}{6})\), then rotate \(\frac{11\pi}{6}\) (or \(330^{\circ}\)) from the positive \(x\) - axis and move 6 units from the origin in that direction.