Question

find an angle between 0 and 2π that is coterminal with the given angle. - \frac{4π}{3}

Explanation:

Step1: Recall coterminal - angle formula

Coterminal angles are found by adding or subtracting multiples of \(2\pi\) to the given angle. Let the given angle be \(\theta=-\frac{4\pi}{3}\), and we want to find \(n\) (an integer) such that \(0\leq\theta + 2n\pi\leq2\pi\).

Step2: Add \(2\pi\) to the given angle

We add \(2\pi\) (where \(n = 1\)) to \(-\frac{4\pi}{3}\). Since \(2\pi=\frac{6\pi}{3}\), then \(-\frac{4\pi}{3}+2\pi=-\frac{4\pi}{3}+\frac{6\pi}{3}\).

Step3: Calculate the sum

\(-\frac{4\pi}{3}+\frac{6\pi}{3}=\frac{- 4\pi+6\pi}{3}=\frac{2\pi}{3}\), and \(0<\frac{2\pi}{3}<2\pi\).

Answer:

\(\frac{2\pi}{3}\)