Question

find the reference angle for each value of $\theta$. give your answers as reduced fractions involving $\pi$.
(a) if $\theta=\frac{2\pi}{3}$, the reference angle is
(b) if $\theta=\frac{3\pi}{4}$, the reference angle is
(c) if $\theta=-\frac{4\pi}{3}$, the reference angle is
(d) if $\theta=\frac{13\pi}{6}$, the reference angle is

Explanation:

Step1: Determine the quadrant for \(\theta=\frac{2\pi}{3}\)

\(\frac{\pi}{2}<\frac{2\pi}{3}<\pi\), so it is in the second quadrant.
The reference angle formula for the second quadrant is \(\pi - \theta\).
\(\pi-\frac{2\pi}{3}=\frac{3\pi - 2\pi}{3}=\frac{\pi}{3}\)

Step2: Determine the quadrant for \(\theta=\frac{3\pi}{4}\)

\(\frac{\pi}{2}<\frac{3\pi}{4}<\pi\), so it is in the second quadrant.
Using the formula \(\pi - \theta\)
\(\pi-\frac{3\pi}{4}=\frac{4\pi - 3\pi}{4}=\frac{\pi}{4}\)

Step3: Determine the coterminal positive angle for \(\theta =-\frac{4\pi}{3}\)

Add \(2\pi\) to get a positive coterminal angle: \(-\frac{4\pi}{3}+2\pi=-\frac{4\pi}{3}+\frac{6\pi}{3}=\frac{2\pi}{3}\)
\(\frac{\pi}{2}<\frac{2\pi}{3}<\pi\), so it is in the second quadrant.
Using the formula \(\pi - \theta\)
\(\pi-\frac{2\pi}{3}=\frac{\pi}{3}\)

Step4: Determine the coterminal angle for \(\theta=\frac{13\pi}{6}\)

Subtract \(2\pi\): \(\frac{13\pi}{6}-2\pi=\frac{13\pi}{6}-\frac{12\pi}{6}=\frac{\pi}{6}\)
\(0<\frac{\pi}{6}<\frac{\pi}{2}\), so the reference angle is itself \(\frac{\pi}{6}\)

Answer:

(a) \(\frac{\pi}{3}\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{3}\)
(d) \(\frac{\pi}{6}\)