Question

1. $-\frac{11pi}{4}$

Explanation:

Step1: Add multiples of \(2\pi\)

To find a coterminal angle, we add \(2\pi\) (or \(\frac{8\pi}{4}\)) to \(-\frac{11\pi}{4}\) until we get an angle between \(0\) and \(2\pi\). First addition: \(-\frac{11\pi}{4}+\frac{8\pi}{4}=-\frac{3\pi}{4}\) (still negative). Second addition: \(-\frac{3\pi}{4}+\frac{8\pi}{4}=\frac{5\pi}{4}\).

Step2: Determine quadrant

\(\frac{5\pi}{4}\) is between \(\pi\) (\(\frac{4\pi}{4}\)) and \(\frac{3\pi}{2}\) (\(\frac{6\pi}{4}\)), so it's in the third quadrant.

Step3: Find reference angle

For an angle \(\theta\) in the third quadrant, the reference angle is \(\theta - \pi\). So \(\frac{5\pi}{4}-\pi=\frac{5\pi}{4}-\frac{4\pi}{4}=\frac{\pi}{4}\).

Answer:

Coterminal angle (between \(0\) and \(2\pi\)): \(\frac{5\pi}{4}\), Quadrant: Third, Reference angle: \(\frac{\pi}{4}\)