To find the exact value of [tex]\(\csc \frac{5\pi}{6}\)[/tex], we break it down into a few understandable steps related to trigonometric identities and properties of the unit circle.
### Step 1: Identify the angle in a more familiar form
The given angle is [tex]\(\frac{5\pi}{6}\)[/tex]. This angle is in radians and it lies in the second quadrant of the unit circle.
### Step 2: Find the reference angle
The reference angle for [tex]\(\frac{5\pi}{6}\)[/tex] is the angle it makes with the x-axis. For angles in the second quadrant, the reference angle can be found by subtracting the given angle from [tex]\(\pi\)[/tex]:
[tex]\[
\text{Reference Angle} = \pi - \frac{5\pi}{6} = \frac{\pi}{6}
\][/tex]
### Step 3: Use the sine function
To find the cosecant of an angle, we first need the sine of the angle. Recall that [tex]\(\csc \theta = \frac{1}{\sin \theta}\)[/tex].
For angles in the second quadrant, the sine value is positive.
Using the reference angle [tex]\(\frac{\pi}{6}\)[/tex]:
[tex]\[
\sin \frac{5\pi}{6} = \sin \frac{\pi}{6} = \frac{1}{2}
\][/tex]
### Step 4: Calculate the cosecant
Now, using the definition of cosecant:
[tex]\[
\csc \frac{5\pi}{6} = \frac{1}{\sin \frac{5\pi}{6}} = \frac{1}{\frac{1}{2}} = 2
\][/tex]
Therefore, the exact value of [tex]\(\csc \frac{5\pi}{6}\)[/tex] is:
[tex]\[
\boxed{2}
\][/tex]
