To find the exact value of [tex]\(\csc\left(\frac{14\pi}{3}\right)\)[/tex], let's go through the steps:
1. Simplify the Angle:
First, we need to simplify the angle [tex]\(\frac{14\pi}{3}\)[/tex]. We do this by reducing it modulo [tex]\(2\pi\)[/tex], as the cosecant function [tex]\(\csc(\theta)\)[/tex] is periodic with a period of [tex]\(2\pi\)[/tex].
[tex]\[
\frac{14\pi}{3} \quad \text{modulo} \quad 2\pi
\][/tex]
[tex]\[
2\pi = \frac{6\pi}{3}
\][/tex]
Now,
[tex]\[
\frac{14\pi}{3} = 4 \cdot \frac{2\pi}{3} + \frac{2\pi}{3}
\][/tex]
By subtracting integer multiples of [tex]\(2\pi\)[/tex] from [tex]\(\frac{14\pi}{3}\)[/tex], we find that:
[tex]\[
\frac{14\pi}{3} \equiv \frac{14\pi}{3} - 4 \cdot 2\pi = \frac{14\pi}{3} - \frac{24\pi}{3} = \frac{14\pi - 24\pi}{3} = -\frac{10\pi}{3}
\][/tex]
Since [tex]\(-\frac{10\pi}{3}\)[/tex] is not within the first [tex]\(2\pi\)[/tex] interval, we can equivalently write the simplified angle as:
[tex]\[
14\pi/3 \equiv 2\pi/3 \quad (\text{mod } 2\pi)
\][/tex]
2. Determine [tex]\(\csc(\theta)\)[/tex]:
Next, let's find [tex]\(\csc(\frac{2\pi}{3})\)[/tex]. Recall that:
[tex]\[
\csc(\theta) = \frac{1}{\sin(\theta)}
\][/tex]
Now, evaluate [tex]\(\sin(\frac{2\pi}{3})\)[/tex]:
[tex]\(\frac{2\pi}{3}\)[/tex] points to an angle in the second quadrant, where sine is positive. The reference angle in this case is [tex]\(\pi - \frac{2\pi}{3} = \frac{\pi}{3}\)[/tex].
[tex]\[
\sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
\][/tex]
Therefore:
[tex]\[
\csc\left(\frac{2\pi}{3}\right) = \frac{1}{\sin\left(\frac{2\pi}{3}\right)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
\][/tex]
Hence, the exact value of [tex]\(\csc\left(\frac{14\pi}{3}\right)\)[/tex] is:
[tex]\[
\frac{2\sqrt{3}}{3}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{2\sqrt{3}}{3}}
\][/tex]
