To find the exact value of [tex]\(\csc \frac{5\pi}{6}\)[/tex], we can follow these steps:
1. Understand the Angle:
[tex]\(\frac{5\pi}{6}\)[/tex] is an angle in radians. Converting it to degrees, we get:
[tex]\[
\frac{5\pi}{6} \times \frac{180^\circ}{\pi} = 150^\circ
\][/tex]
2. Use the Unit Circle:
On the unit circle, the angle [tex]\(150^\circ\)[/tex] lies in the second quadrant. The reference angle for [tex]\(150^\circ\)[/tex] is:
[tex]\[
180^\circ - 150^\circ = 30^\circ
\][/tex]
The sine of an angle in the second quadrant is positive.
3. Sine of the Reference Angle:
The sine of [tex]\(30^\circ\)[/tex] (or [tex]\(\frac{\pi}{6}\)[/tex] radians) is:
[tex]\[
\sin 30^\circ = \frac{1}{2}
\][/tex]
4. Cosecant is the Reciprocal of Sine:
The cosecant function [tex]\(\csc\theta\)[/tex] is defined as the reciprocal of the sine function [tex]\(\sin\theta\)[/tex]:
[tex]\[
\csc\theta = \frac{1}{\sin\theta}
\][/tex]
5. Calculate Cosecant:
Therefore, the exact value of [tex]\(\csc 150^\circ\)[/tex] or [tex]\(\csc \frac{5\pi}{6}\)[/tex] is the reciprocal of [tex]\(\sin 150^\circ\)[/tex]:
[tex]\[
\csc \frac{5\pi}{6} = \frac{1}{\sin \frac{5\pi}{6}} = \frac{1}{\frac{1}{2}} = 2
\][/tex]
Thus, the exact value of [tex]\(\csc \frac{5\pi}{6}\)[/tex] is [tex]\(2\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{2}
\][/tex]
