Question

find the reference angle for $\frac{5\pi}{4}$. the reference angle is $\square$. (simplify your answer. use integers or fractions for any numbers in the expression. type your answer in radians.)

Explanation:

Step1: Determine the quadrant of \(\frac{5\pi}{4}\)

The angle \(\frac{5\pi}{4}\) is greater than \(\pi\) (\(\frac{4\pi}{4}\)) and less than \(\frac{3\pi}{2}\) (\(\frac{6\pi}{4}\)), so it lies in the third quadrant.

Step2: Use the formula for reference angle in third quadrant

For an angle \(\theta\) in the third quadrant (\(\pi<\theta<\frac{3\pi}{2}\)), the reference angle \(r\) is given by \(r = \theta-\pi\).
Substitute \(\theta=\frac{5\pi}{4}\) into the formula:
\(r=\frac{5\pi}{4}-\pi\)
Since \(\pi = \frac{4\pi}{4}\), we have \(r=\frac{5\pi}{4}-\frac{4\pi}{4}=\frac{\pi}{4}\)

Answer:

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