Answer
$\dfrac{\pi}{4}$
Work Step by Step
$\dfrac{23\pi}{4}$ is close to $\dfrac{24\pi}{4}=6\pi$.
This means that this angle is between $5.5\pi$ and $6\pi$.
RECALL:
An angle $\theta$, where $5.5\pi \lt \theta \lt 6\pi$, is coterminal with:
$\theta-4\pi$
Thus, the given angle is coterminal with:
$=\dfrac{23\pi}{4}-4\pi=\dfrac{23\pi}{4} - \dfrac{16\pi}{4} = \dfrac{7\pi}{4}$
$\dfrac{7\pi}{4}$ is coterminal with $\dfrac{23\pi}{4}$.
$\dfrac{7\pi}{4}$ is in Quadrant IV so $\dfrac{23\pi}{4}$ is also in Quadrant IV.
RECALL:
The following are the means on how to find the reference angle of an angle $0 \leq \theta \lt 2\pi$ based on its position:
(1) Quadrant I: $\theta$ (itself)
(2) Quadrant II: $\pi-\theta$
(3) Quadrant III: $\theta - \pi$
(4) Quadrant IV: $2\pi - \theta$
Use formula (4) above to obtain:
reference angle of $\dfrac{23\pi}{4}$ = reference angle of $\dfrac{7\pi}{4}$, which is
$\\2\pi - \dfrac{7\pi}{4}
\\=\dfrac{8\pi}{4} - \dfrac{7\pi}{4}
\\=\dfrac{\pi}{4}$
