To determine the exact value of [tex]\(\cos (\pi + x)\)[/tex], we will use the angle addition formula for cosine, specifically for the scenario where we add [tex]\(\pi\)[/tex] to another angle.
Given:
[tex]\[ \cos (\pi + x) \][/tex]
The angle addition formula for cosine is:
[tex]\[ \cos (a + b) = \cos a \cos b - \sin a \sin b \][/tex]
For our specific case, we have [tex]\(\cos (\pi + x)\)[/tex]. We substitute [tex]\(a = \pi\)[/tex] and [tex]\(b = x\)[/tex]:
[tex]\[ \cos (\pi + x) = \cos \pi \cos x - \sin \pi \sin x \][/tex]
Now, let's recall the values of [tex]\(\cos \pi\)[/tex] and [tex]\(\sin \pi\)[/tex]:
- [tex]\(\cos \pi = -1\)[/tex]
- [tex]\(\sin \pi = 0\)[/tex]
Substituting these values in, we get:
[tex]\[ \cos (\pi + x) = (-1) \cos x - (0) \sin x \][/tex]
[tex]\[ \cos (\pi + x) = -\cos x \][/tex]
Therefore, [tex]\(\cos (\pi + x)\)[/tex] is simply [tex]\(-\cos x\)[/tex].
Now, we need to find the exact value given the provided choices. From the computation:
[tex]\[ \cos (\pi + x) = -\cos x \][/tex]
To match one of the provided choices:
[tex]\[ -\cos x = -\frac{1}{2} \][/tex]
Thus, the exact value is:
[tex]\[ -\frac{1}{2} \][/tex]
So, the correct answer, from the provided choices, is:
[tex]\[ -\frac{1}{2} \][/tex]
