Question

find the measures of two angles, one positive and one negative, that are coterminal with the given angle.
\\( \frac { 2 \pi } { 7 } \\)
\\( \frac { 9 \pi } { 7 } ; - \frac { 12 \pi } { 7 } \\)
\\( \frac { 12 \pi } { 7 } ; - \frac { 16 \pi } { 7 } \\)
\\( \frac { 9 \pi } { 7 } ; - \frac { 9 \pi } { 7 } \\)
\\( \frac { 16 \pi } { 7 } ; - \frac { 12 \pi } { 7 } \\)

Explanation:

Step1: Recall the formula for coterminal angles

Coterminal angles with an angle \(\theta\) are given by \(\theta + 2k\pi\), where \(k\in\mathbb{Z}\) ( \(k\) is an integer). For \(\theta=\frac{2\pi}{7}\), when \(k = 1\):

$$ \frac{2\pi}{7}+2\pi=\frac{2\pi + 14\pi}{7}=\frac{16\pi}{7} $$

Step2: Find the negative coterminal angle

When \(k=- 1\):

$$ \frac{2\pi}{7}-2\pi=\frac{2\pi-14\pi}{7}=-\frac{12\pi}{7} $$

Answer:

\(\frac{16\pi}{7};-\frac{12\pi}{7}\) (the fourth option)