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: Find positive coterminal angle

To find a positive coterminal angle with \(\frac{2\pi}{7}\), add \(2\pi\) (since \(2\pi\) is a full - rotation in radians).

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

Step2: Find negative coterminal angle

To find a negative coterminal angle with \(\frac{2\pi}{7}\), subtract \(2\pi\).

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

Answer:

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