Question

find the measure of the reference angle for a ( \frac{31pi}{18} ) radian angle. give the answer in simplest form. input box radians

Explanation:

Step1: Determine the quadrant of \(\frac{31\pi}{18}\)

We know that \(2\pi = \frac{36\pi}{18}\), and \(\pi=\frac{18\pi}{18}\), \(\frac{3\pi}{2}=\frac{27\pi}{18}\). Since \(\frac{27\pi}{18}<\frac{31\pi}{18}<\frac{36\pi}{18}\), the angle \(\frac{31\pi}{18}\) is in the fourth quadrant.

Step2: Calculate the reference angle

For an angle \(\theta\) in the fourth quadrant, the reference angle \(\theta_{ref}\) is given by \(2\pi-\theta\). So we substitute \(\theta = \frac{31\pi}{18}\) into the formula:
\(\theta_{ref}=2\pi-\frac{31\pi}{18}\)
We know that \(2\pi=\frac{36\pi}{18}\), so \(\theta_{ref}=\frac{36\pi}{18}-\frac{31\pi}{18}=\frac{36\pi - 31\pi}{18}=\frac{5\pi}{18}\)

Answer:

\(\frac{5\pi}{18}\)