Question

which angle is coterminal with \frac{7\pi}{6}? \frac{14\pi}{6} -\frac{\pi}{6} -\frac{5\pi}{6}

Explanation:

Step1: Recall the formula for coterminal angles

Coterminal angles are given by \(\theta + 2k\pi\), where \(k\in\mathbb{Z}\).

Step2: Check each option

• For \(\frac{14\pi}{6}\): \(\frac{7\pi}{6}+2k\pi=\frac{7\pi}{6}+ 2\pi=\frac{7\pi + 12\pi}{6}=\frac{19\pi}{6}

eq\frac{14\pi}{6}\) (when \(k = 1\)).

• For \(-\frac{\pi}{6}\): \(\frac{7\pi}{6}+2k\pi\). Let \(k=-1\), then \(\frac{7\pi}{6}-2\pi=\frac{7\pi - 12\pi}{6}=-\frac{5\pi}{6}

eq-\frac{\pi}{6}\).

• For \(-\frac{5\pi}{6}\): Let \(k=-1\) in \(\theta+2k\pi\) (\(\theta=\frac{7\pi}{6}\)), \(\frac{7\pi}{6}+2(- 1)\pi=\frac{7\pi}{6}-2\pi=\frac{7\pi - 12\pi}{6}=-\frac{5\pi}{6}\).

Answer:

\(-\frac{5\pi}{6}\)