To determine the reference angle for [tex]\(\frac{5\pi}{3}\)[/tex], follow these steps:
1. Recognize which quadrant the angle [tex]\(\frac{5\pi}{3}\)[/tex] lies in. Angles in the range [tex]\(\frac{3\pi}{2} < \theta < 2\pi\)[/tex] are in the fourth quadrant because [tex]\(2\pi = \frac{6\pi}{3}\)[/tex] and [tex]\(\frac{5\pi}{3}\)[/tex] is just slightly less than [tex]\(\frac{6\pi}{3}\)[/tex].
2. In the fourth quadrant, the reference angle [tex]\(\theta_{\text{ref}}\)[/tex] can be found by subtracting the given angle from [tex]\(2\pi\)[/tex]:
[tex]\[
\theta_{\text{ref}} = 2\pi - \frac{5\pi}{3}
\][/tex]
3. Perform the subtraction:
[tex]\[
\theta_{\text{ref}} = 2\pi - \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3}
\][/tex]
Thus, the reference angle for [tex]\(\frac{5\pi}{3}\)[/tex] is [tex]\(\boxed{\frac{\pi}{3}}\)[/tex].
