Question

find the measures of two angles, one positive and one negative, that are coterminal with the given angle.
\\( \frac { \pi } { 6 } \\)
\\( \frac { \pi } { 6 } + 360 ^ { \circ } ; \frac { \pi } { 6 } - 360 ^ { \circ } \\)
\\( \frac { 13 \pi } { 6 } ; - \frac { 11 \pi } { 6 } \\)
\\( \frac { 13 \pi } { 6 } ; - \frac { \pi } { 6 } \\)
\\( \frac { 7 \pi } { 6 } ; - \frac { 5 \pi } { 6 } \\)

Explanation:

Step1: Recall the formula for coterminal angles

Coterminal angles can be found by adding or subtracting \(2\pi\) (since \(2\pi = 360^{\circ}\) in radians). For an angle \(\theta\), coterminal angles are \(\theta + 2k\pi\) and \(\theta-2k\pi\), where \(k = 1,2,\cdots\)

Step2: Calculate positive coterminal angle

Let \(\theta=\frac{\pi}{6}\) and \(k = 1\). Then \(\frac{\pi}{6}+2\pi=\frac{\pi + 12\pi}{6}=\frac{13\pi}{6}\)

Step3: Calculate negative coterminal angle

Let \(\theta=\frac{\pi}{6}\) and \(k = 1\). Then \(\frac{\pi}{6}-2\pi=\frac{\pi- 12\pi}{6}=-\frac{11\pi}{6}\)

Answer:

\(\frac{13\pi}{6};-\frac{11\pi}{6}\) (the second option)