Certainly! Let's determine the value of [tex]\(\cos\left(\frac{3\pi}{4}\right)\)[/tex].
### Step-by-Step Solution:
1. Understand the Angle:
- The angle [tex]\(\frac{3\pi}{4}\)[/tex] is in radians.
- Converting this to degrees, we get:
[tex]\[
\frac{3\pi}{4} \times \frac{180^\circ}{\pi} = 135^\circ
\][/tex]
2. Determine the Quadrant:
- The angle [tex]\(135^\circ\)[/tex] is in the second quadrant.
- Recall that in the second quadrant, the cosine function is negative.
3. Reference Angle:
- The reference angle for [tex]\(135^\circ\)[/tex] is:
[tex]\[
180^\circ - 135^\circ = 45^\circ
\][/tex]
4. Cosine of the Reference Angle:
- The cosine of [tex]\(45^\circ\)[/tex] (or [tex]\(\frac{\pi}{4}\)[/tex] radians) is:
[tex]\[
\cos 45^\circ = \frac{\sqrt{2}}{2}
\][/tex]
5. Applying the Sign:
- Since [tex]\(135^\circ\)[/tex] is in the second quadrant, where cosine values are negative, we have:
[tex]\[
\cos 135^\circ = -\cos 45^\circ = -\frac{\sqrt{2}}{2}
\][/tex]
Hence, the value of [tex]\(\cos\left(\frac{3\pi}{4}\right)\)[/tex] is:
[tex]\[
-\frac{\sqrt{2}}{2}
\][/tex]
### Final Answer:
D. [tex]\(-\frac{\sqrt{2}}{2}\)[/tex]
