Answer
a. $\displaystyle \frac{\pi}{6}$
b. $\displaystyle \frac{\pi}{9}$
c. $\displaystyle \frac{2\pi}{7}$
Work Step by Step
The reference angle $\overline{\theta}$ associated with an angle $\theta$
is the acute angle formed by the terminal side of $\theta$ and the x-axis.
a.
$\displaystyle \frac{5\pi}{6}$ has terminal side in quadrant II.
The acute angle it forms with the x-axis is
$\displaystyle \overline{\theta}=\pi-\frac{5\pi}{6}=\frac{\pi}{6}.$
b.
$\displaystyle \frac{10\pi}{9}$ has terminal side in quadrant III.
The acute angle it forms with the x-axis is
$\displaystyle \overline{\theta}=\frac{10\pi}{9} -\displaystyle \pi=\frac{\pi}{9}.$
c.
$\displaystyle \frac{23\pi}{7}=2\pi+\frac{9\pi}{7}$ has terminal side in quadrant III.
The acute angle it forms with the x-axis is
$\displaystyle \overline{\theta}=\frac{9\pi}{7} -\displaystyle \pi=\frac{2\pi}{7}.$
