Question

1.) rotating in the positive direction, label each common angle in radians on the circle below. begin with 0 and end with 2π.
2.) rotating in the negative direction, label the common angles in radians. include every angle from -2π to 0.

Explanation:

Step1: Recall the relationship between positive and negative angles

Angles in the negative direction are equivalent to \(2\pi\) minus the corresponding positive - angle. For example, if we have a positive angle \(\theta\), the negative - equivalent angle \(\alpha=-\theta + 2k\pi\) (in the range \(-2\pi\leqslant\alpha\leqslant0\), \(k = 0\) or \(k = 1\)).

Step2: List the common angles

The common angles (in positive direction) we usually consider are \(0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2},\frac{2\pi}{3},\frac{3\pi}{4},\frac{5\pi}{6},\pi,\frac{7\pi}{6},\frac{5\pi}{4},\frac{4\pi}{3},\frac{3\pi}{2},\frac{5\pi}{3},\frac{7\pi}{4},\frac{11\pi}{6},2\pi\).
For \(\theta=\frac{\pi}{6}\), the negative - equivalent angle \(\alpha=- \frac{11\pi}{6}\) (since \(-\frac{\pi}{6}+2\pi=\frac{11\pi}{6}\) is positive, \(-\frac{\pi}{6}\) is not in the range \(-2\pi\leqslant\alpha\leqslant0\), and \(-\frac{\pi}{6}- 2\pi=-\frac{13\pi}{6}\) is not a common angle. Using \(\alpha = 2\pi-\frac{\pi}{6}-2\pi=-\frac{11\pi}{6}\)).
For \(\theta=\frac{\pi}{4}\), \(\alpha=-\frac{7\pi}{4}\) (because \(2\pi-\frac{\pi}{4}-2\pi =-\frac{7\pi}{4}\)).
For \(\theta=\frac{\pi}{3}\), \(\alpha=-\frac{5\pi}{3}\) (since \(2\pi-\frac{\pi}{3}-2\pi=-\frac{5\pi}{3}\)).
For \(\theta=\frac{\pi}{2}\), \(\alpha=-\frac{3\pi}{2}\) (because \(2\pi-\frac{\pi}{2}-2\pi=-\frac{3\pi}{2}\)).
For \(\theta=\frac{2\pi}{3}\), \(\alpha=-\frac{4\pi}{3}\) (since \(2\pi-\frac{2\pi}{3}-2\pi=-\frac{4\pi}{3}\)).
For \(\theta=\frac{3\pi}{4}\), \(\alpha=-\frac{5\pi}{4}\) (because \(2\pi-\frac{3\pi}{4}-2\pi=-\frac{5\pi}{4}\)).
For \(\theta=\frac{5\pi}{6}\), \(\alpha=-\frac{7\pi}{6}\) (since \(2\pi-\frac{5\pi}{6}-2\pi=-\frac{7\pi}{6}\)).
For \(\theta=\pi\), \(\alpha=-\pi\).

Answer:

Starting from the positive \(x -\)axis (going clock - wise): \(0,-\frac{\pi}{6},-\frac{\pi}{4},-\frac{\pi}{3},-\frac{\pi}{2},-\frac{2\pi}{3},-\frac{3\pi}{4},-\frac{5\pi}{6},-\pi,-\frac{7\pi}{6},-\frac{5\pi}{4},-\frac{4\pi}{3},-\frac{3\pi}{2},-\frac{5\pi}{3},-\frac{7\pi}{4},-\frac{11\pi}{6},- 2\pi\)