Discover a world of knowledge and get your questions answered at IDNLearn.com. Join our platform to receive prompt and accurate responses from experienced professionals in various fields.

(05.04 MC)

What is the exact value of [tex]\csc \frac{2 \pi}{3}[/tex]?

A. [tex]-2[/tex]

B. [tex]\frac{2 \sqrt{3}}{3}[/tex]

C. [tex]-\frac{\sqrt{3}}{3}[/tex]

D. [tex]\frac{3 \sqrt{2}}{2}[/tex]


Sagot :

To determine the exact value of \(\csc \frac{2\pi}{3}\), we first recall that cosecant is the reciprocal of the sine function. Therefore, \(\csc x = \frac{1}{\sin x}\).

Let's find \(\sin \frac{2\pi}{3}\):

1. \(\frac{2\pi}{3}\) lies in the second quadrant. The reference angle for \(\frac{2\pi}{3}\) is \(\pi - \frac{2\pi}{3} = \frac{\pi}{3}\).
2. In the second quadrant, sine is positive. Recall that \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\).
3. Therefore, \(\sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}\).

Now, taking the reciprocal to find \(\csc \frac{2\pi}{3}\):

[tex]\[ \csc \frac{2\pi}{3} = \frac{1}{\sin \frac{2\pi}{3}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \][/tex]

We can rationalize the denominator:

[tex]\[ \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \][/tex]

So, the exact value of \(\csc \frac{2\pi}{3}\) is \(\frac{2\sqrt{3}}{3}\).

The correct answer is [tex]\(\frac{2 \sqrt{3}}{3}\)[/tex].