Question

find the reference angle for $-\frac{2pi}{3}$. radians

Explanation:

Step1: Determine the positive - equivalent angle

Since the given angle is \(-\frac{2\pi}{3}\), we add \(2\pi\) to get a positive - equivalent angle. \(-\frac{2\pi}{3}+2\pi=-\frac{2\pi}{3}+\frac{6\pi}{3}=\frac{4\pi}{3}\).

Step2: Identify the quadrant

The angle \(\frac{4\pi}{3}\) lies in the third quadrant because \(\pi<\frac{4\pi}{3}<\frac{3\pi}{2}\).

Step3: Calculate the reference angle

For an angle \(\theta\) in the third quadrant, the reference angle \(\theta_{r}=\theta - \pi\). So, \(\theta_{r}=\frac{4\pi}{3}-\pi=\frac{4\pi - 3\pi}{3}=\frac{\pi}{3}\).

Answer:

\(\frac{\pi}{3}\)