Question

evaluate without a calculator: csc\frac{5\pi}{3} type + or - \frac{\sqrt{}}{}

Explanation:

Step1: Convert csc to sin

Recall that \(\csc\theta=\frac{1}{\sin\theta}\), so \(\csc\frac{5\pi}{3}=\frac{1}{\sin\frac{5\pi}{3}}\).

Step2: Find the reference angle

The angle \(\frac{5\pi}{3}\) is in the fourth - quadrant. The reference angle \(\theta_{r}=2\pi-\frac{5\pi}{3}=\frac{\pi}{3}\). And in the fourth - quadrant, \(\sin\theta\lt0\). So \(\sin\frac{5\pi}{3}=-\sin\frac{\pi}{3}\).

Step3: Evaluate \(\sin\frac{\pi}{3}\)

Since \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\), then \(\sin\frac{5\pi}{3}=-\frac{\sqrt{3}}{2}\).

Step4: Calculate \(\csc\frac{5\pi}{3}\)

\(\csc\frac{5\pi}{3}=\frac{1}{\sin\frac{5\pi}{3}}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}\)

Answer:

\(-\frac{2\sqrt{3}}{3}\)