To find the exact value of [tex]\(\cot \frac{7 \pi}{4}\)[/tex], let's first understand its definition and the calculation involved.
1. Understanding the Angle:
The angle [tex]\(\frac{7 \pi}{4}\)[/tex] radians can be converted to degrees:
[tex]\[
\frac{7 \pi}{4} \times \frac{180^\circ}{\pi} = 315^\circ
\][/tex]
The angle [tex]\(315^\circ\)[/tex] lies in the fourth quadrant of the unit circle.
2. Using Trigonometric Definitions:
- The cotangent function is defined as the reciprocal of the tangent function:
[tex]\[
\cot \theta = \frac{1}{\tan \theta}
\][/tex]
3. Finding [tex]\(\tan \frac{7 \pi}{4}\)[/tex]:
- In the fourth quadrant, the tangent of an angle is negative, and for the angle [tex]\(315^\circ\)[/tex] (or [tex]\(\frac{7 \pi}{4}\)[/tex]), it is known that:
[tex]\[
\tan 315^\circ = \tan(360^\circ - 45^\circ) = -\tan 45^\circ = -1
\][/tex]
4. Calculating [tex]\(\cot \frac{7 \pi}{4}\)[/tex]:
- Using the fact that [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex], the cotangent of [tex]\(\frac{7 \pi}{4}\)[/tex] can be determined:
[tex]\[
\cot \frac{7 \pi}{4} = \frac{1}{\tan \frac{7 \pi}{4}} = \frac{1}{-1} = -1
\][/tex]
So, the exact value of [tex]\(\cot \frac{7 \pi}{4}\)[/tex] is [tex]\(\boxed{-1}\)[/tex].
