Find the value of:

[tex]\[
\cos \left(\frac{3 \pi}{4}\right) =
\][/tex]

A. [tex]\(\frac{\sqrt{2}}{2}\)[/tex]

B. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]

C. [tex]\(\frac{1}{2}\)[/tex]

D. [tex]\(-\frac{\sqrt{2}}{2}\)[/tex]



Answer :

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]