Question

find the reference angle for the angle $-\frac{21pi}{4}$. the reference angle is (type your answer in radians. type an integer or a simplified fraction.)

Explanation:

Step1: Make the angle positive

Add \(2\pi\) multiple - times until the angle is positive. Since \(2\pi=\frac{8\pi}{4}\), \(-\frac{21\pi}{4}+6\times\frac{8\pi}{4}=-\frac{21\pi}{4}+\frac{48\pi}{4}=\frac{27\pi}{4}\).

Step2: Reduce the angle to one full - rotation

\(\frac{27\pi}{4}-6\pi=\frac{27\pi}{4}-\frac{24\pi}{4}=\frac{3\pi}{4}\).

Step3: Determine the reference angle

The angle \(\frac{3\pi}{4}\) is in the second quadrant. The reference angle \(\theta_{r}\) for an angle \(\theta\) in the second quadrant is \(\pi-\theta\). So, \(\theta_{r}=\pi - \frac{3\pi}{4}=\frac{\pi}{4}\).

Answer:

\(\frac{\pi}{4}\)