Question

find the measures of two angles, one positive and one negative, that are coterminal with the given angle.
\\( \frac { 2 \pi } { 3 } \\)
\\( \frac { 5 \pi } { 3 } ; - \frac { 5 \pi } { 3 } \\)
\\( \frac { 4 \pi } { 3 } ; - \frac { 8 \pi } { 3 } \\)
\\( \frac { 8 \pi } { 3 } ; - \frac { 4 \pi } { 3 } \\)
\\( \frac { 5 \pi } { 3 } ; - \frac { 4 \pi } { 3 } \\)

Explanation:

Step1: Recall the formula for coterminal angles

Coterminal angles of an angle \(\theta\) are given by \(\theta + 2k\pi\), where \(k\in\mathbb{Z}\).
For \(\theta=\frac{2\pi}{3}\):

• When \(k = 1\):

\(\frac{2\pi}{3}+2\pi=\frac{2\pi + 6\pi}{3}=\frac{8\pi}{3}\)

• When \(k=-1\):

\(\frac{2\pi}{3}-2\pi=\frac{2\pi-6\pi}{3}=-\frac{4\pi}{3}\)

Answer:

\(\frac{8\pi}{3};-\frac{4\pi}{3}\) (corresponding to the third option)