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})