find the exact values below. if applicable, click on \undefined\.\n\ncsc\\(\\frac { 2 \\pi } { 3 }\\) =…

Explanation:

Step1: Find the value of (\csc\frac{2\pi}{3})

Recall that (\csc\theta=\frac{1}{\sin\theta}). First, find (\sin\frac{2\pi}{3}). We know that (\sin\frac{2\pi}{3}=\sin(\pi - \frac{\pi}{3})). Using the identity (\sin(\pi-\alpha)=\sin\alpha), so (\sin\frac{2\pi}{3}=\sin\frac{\pi}{3}). And (\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}). Then (\csc\frac{2\pi}{3}=\frac{1}{\sin\frac{2\pi}{3}}=\frac{1}{\frac{\sqrt{3}}{2}}=\frac{2\sqrt{3}}{3}).

Step2: Find the value of (\cot\frac{2\pi}{3})

Recall that (\cot\theta=\frac{\cos\theta}{\sin\theta}). Find (\cos\frac{2\pi}{3}) and (\sin\frac{2\pi}{3}). (\cos\frac{2\pi}{3}=\cos(\pi - \frac{\pi}{3})). Using the identity (\cos(\pi-\alpha)=-\cos\alpha), so (\cos\frac{2\pi}{3}=-\cos\frac{\pi}{3}=-\frac{1}{2}). (\sin\frac{2\pi}{3}=\frac{\sqrt{3}}{2}). Then (\cot\frac{2\pi}{3}=\frac{\cos\frac{2\pi}{3}}{\sin\frac{2\pi}{3}}=\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}=-\frac{\sqrt{3}}{3}).

Answer:

(\csc\frac{2\pi}{3}=\frac{2\sqrt{3}}{3}), (\cot\frac{2\pi}{3}=-\frac{\sqrt{3}}{3})