To solve the problem of finding the exact value of [tex]\(\sec \frac{3\pi}{4}\)[/tex], let's proceed step by step:
1. Recognize the definition:
The secant function is the reciprocal of the cosine function. Therefore,
[tex]\[
\sec \theta = \frac{1}{\cos \theta}
\][/tex]
2. Identify [tex]\(\theta\)[/tex]:
Here, [tex]\(\theta = \frac{3\pi}{4}\)[/tex].
3. Determine the cosine of the angle:
The angle [tex]\(\frac{3\pi}{4}\)[/tex] is in the second quadrant, where the cosine function is negative. We can use the unit circle or known values of trigonometric functions to find:
[tex]\[
\cos \left(\frac{3\pi}{4}\right) = -\cos \left(\frac{\pi}{4}\right)
\][/tex]
Since [tex]\(\cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)[/tex], we have:
[tex]\[
\cos \left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\][/tex]
4. Calculate the secant:
Using the definition of secant, we get:
[tex]\[
\sec \left(\frac{3\pi}{4}\right) = \frac{1}{\cos \left(\frac{3\pi}{4}\right)} = \frac{1}{-\frac{\sqrt{2}}{2}}
\][/tex]
5. Simplify the expression:
Simplifying the fraction, we get:
[tex]\[
\frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = -\frac{2 \sqrt{2}}{2} = -\sqrt{2}
\][/tex]
Therefore, the exact value of [tex]\(\sec \left(\frac{3\pi}{4}\right)\)[/tex] is:
[tex]\[
-\sqrt{2}
\][/tex]
