Question

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

Explanation:

Step1: Convert csc to sin

We know that \(\csc x=\frac{1}{\sin x}\), 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 \(\theta' = 2\pi-\frac{5\pi}{3}=\frac{\pi}{3}\)
In the fourth quadrant, \(\sin\) is negative. And \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\), so \(\sin\frac{5\pi}{3}=-\frac{\sqrt{3}}{2}\)

Step3: Calculate \(\csc\frac{5\pi}{3}\)

Since \(\csc\frac{5\pi}{3}=\frac{1}{\sin\frac{5\pi}{3}}\), substituting \(\sin\frac{5\pi}{3}=-\frac{\sqrt{3}}{2}\), we get \(\csc\frac{5\pi}{3}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}\)

Answer:

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