To find the value of [tex]\(\cos 225^\circ\)[/tex], let’s work through the problem step by step.
1. Understand the Angle:
The angle given is [tex]\(225^\circ\)[/tex]. This is an angle in the third quadrant of the unit circle, since [tex]\(225^\circ = 180^\circ + 45^\circ\)[/tex].
2. Reference Angle:
The reference angle for [tex]\(225^\circ\)[/tex] is calculated by subtracting [tex]\(180^\circ\)[/tex] (since it's in the third quadrant):
[tex]\[
225^\circ - 180^\circ = 45^\circ
\][/tex]
So, the reference angle is [tex]\(45^\circ\)[/tex].
3. Cosine of Reference Angle:
The cosine of [tex]\(45^\circ\)[/tex] is known:
[tex]\[
\cos 45^\circ = \frac{\sqrt{2}}{2}
\][/tex]
4. Considering the Quadrant:
Cosine values in the third quadrant are negative because both the x-coordinate and y-coordinate are negative in this quadrant. Therefore, the cosine of [tex]\(225^\circ\)[/tex] is:
[tex]\[
\cos 225^\circ = -\frac{\sqrt{2}}{2}
\][/tex]
By breaking down the problem and considering the properties of the unit circle, we find:
[tex]\[
\cos 225^\circ = -\frac{\sqrt{2}}{2}
\][/tex]
Therefore, among the given options, the exact value of [tex]\(\cos 225^\circ\)[/tex] is:
[tex]\[
-\frac{\sqrt{2}}{2}
\][/tex]
