To find the exact value of [tex]\(\cot \frac{5\pi}{4}\)[/tex], we must understand what cotangent represents. Cotangent is the reciprocal of the tangent function.
Given:
[tex]\[
\cot \frac{5\pi}{4} = \frac{1}{\tan \frac{5\pi}{4}}
\][/tex]
First, we need to determine the value of [tex]\(\tan \frac{5\pi}{4}\)[/tex].
1. Reference Angle:
The angle [tex]\(\frac{5\pi}{4}\)[/tex] is in the third quadrant of the unit circle, where both sine and cosine are negative. The reference angle is:
[tex]\[
\pi - \frac{5\pi}{4} = \frac{5\pi}{4} - \pi = \frac{\pi}{4}
\][/tex]
2. Tangent in the Third Quadrant:
Since [tex]\(\frac{5\pi}{4}\)[/tex] is in the third quadrant, where tangent is positive (both sine and cosine are negative, and negative divided by negative is positive):
[tex]\[
\tan \frac{5\pi}{4} = \tan \frac{\pi}{4} = 1
\][/tex]
Therefore:
[tex]\[
\cot \frac{5\pi}{4} = \frac{1}{\tan \frac{5\pi}{4}} = \frac{1}{1} = 1
\][/tex]
Hence,
[tex]\[
\cot \frac{5\pi}{4} = 1
\][/tex]
Thus, the exact value of [tex]\(\cot \frac{5\pi}{4}\)[/tex] is [tex]\(1\)[/tex].
