Question

find the reference angle for the angle $\frac{47pi}{6}$. the reference angle is (type your answer in radians. type an integer or a simplified fraction.)

Explanation:

Step1: Find a coterminal angle.

First, find a coterminal angle of $\frac{47\pi}{6}$ that is between $0$ and $2\pi$. We know that $2\pi=\frac{12\pi}{6}$. Divide $\frac{47\pi}{6}$ by $\frac{12\pi}{6}$: $\frac{47\pi}{6}\div\frac{12\pi}{6}=\frac{47}{12}=3\frac{11}{12}$. So, $\frac{47\pi}{6}= 7\pi+\frac{5\pi}{6}$. Since $7\pi = 3\times2\pi+\pi$, we can rewrite $\frac{47\pi}{6}$ as $3\times2\pi+\pi+\frac{5\pi}{6}$. A coterminal angle in $[0, 2\pi]$ is $\frac{47\pi}{6}- 4\times2\pi=\frac{47\pi}{6}-\frac{48\pi}{6}=-\frac{\pi}{6}+2\pi=\frac{11\pi}{6}$.

Step2: Determine the reference angle.

The angle $\theta=\frac{11\pi}{6}$ is in the fourth - quadrant. The formula for the reference angle $\theta_{r}$ of an angle $\theta$ in the fourth - quadrant is $\theta_{r}=2\pi-\theta$. So, $\theta_{r}=2\pi-\frac{11\pi}{6}=\frac{12\pi - 11\pi}{6}=\frac{\pi}{6}$.

Answer:

$\frac{\pi}{6}$