Question

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csc\\(\frac { 2 \pi } { 3 }\\) =

cot\\(\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:

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