To determine the exact value of [tex]\(\csc\left(\frac{4\pi}{3}\right)\)[/tex], let's follow these steps:
### Step 1: Understand the Argument and its Quadrant
The angle [tex]\(\frac{4\pi}{3}\)[/tex] is in standard position. This angle is measured in radians. First, let's convert [tex]\(\frac{4\pi}{3}\)[/tex] to degrees to understand its position:
[tex]\[
\frac{4\pi}{3} \times \frac{180^\circ}{\pi} = 240^\circ
\][/tex]
So, [tex]\(\frac{4\pi}{3}\)[/tex] radians is equivalent to [tex]\(240^\circ\)[/tex].
### Step 2: Determine the Sine of the Angle
The angle [tex]\(240^\circ\)[/tex] lies in the third quadrant, where sine is negative. The reference angle for [tex]\(240^\circ\)[/tex] is:
[tex]\[
240^\circ - 180^\circ = 60^\circ
\][/tex]
Given that [tex]\(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)[/tex], and since sine is negative in the third quadrant,
[tex]\[
\sin\left(240^\circ\right) = \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}
\][/tex]
### Step 3: Calculate the Cosecant
The cosecant function is the reciprocal of the sine function. Therefore,
[tex]\[
\csc\left(\frac{4\pi}{3}\right) = \frac{1}{\sin\left(\frac{4\pi}{3}\right)}
\][/tex]
Substitute the value of [tex]\(\sin\left(\frac{4\pi}{3}\right)\)[/tex] we found in Step 2:
[tex]\[
\csc\left(\frac{4\pi}{3}\right) = \frac{1}{-\frac{\sqrt{3}}{2}}
\][/tex]
Simplify the reciprocal:
[tex]\[
\csc\left(\frac{4\pi}{3}\right) = -\frac{2}{\sqrt{3}}
\][/tex]
### Step 4: Simplify and Verify the Answer
So, the exact value of [tex]\(\csc\left(\frac{4\pi}{3}\right)\)[/tex] is:
[tex]\[
-\frac{2}{\sqrt{3}}
\][/tex]
Thus, the correct answer is:
[tex]\[
-\frac{2}{\sqrt{3}}
\][/tex]
