Question

2 multiple choice 1 point evaluate without using a calculator by using ratios in a reference triangle. sin \frac{5pi}{3} \frac{sqrt{3}}{2} -\frac{sqrt{3}}{2} \frac{1}{2} -\frac{sqrt{2}}{2}

Explanation:

Step1: Find the reference angle

The angle \(\frac{5\pi}{3}\) is in the fourth - quadrant. The reference angle \(\theta_{r}=2\pi-\frac{5\pi}{3}=\frac{\pi}{3}\).

Step2: Determine the sign of the sine function

In the fourth - quadrant, \(\sin\theta<0\).

Step3: Use the sine value of the reference angle

We know that \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\). Since \(\sin\frac{5\pi}{3}<0\), then \(\sin\frac{5\pi}{3}=-\sin\frac{\pi}{3}\).

Answer:

\(-\frac{\sqrt{3}}{2}\) (the second option)