To determine the value of [tex]\(\cos(3\pi)\)[/tex], let's go through the solution step-by-step:
1. Understanding the angle: We are given the angle [tex]\(3\pi\)[/tex]. Since [tex]\(\pi\)[/tex] radians correspond to 180 degrees, [tex]\(3\pi\)[/tex] radians equal [tex]\(3 \times 180^\circ = 540^\circ\)[/tex].
2. Simplifying the angle: The cosine function is periodic with period [tex]\(2\pi\)[/tex], meaning [tex]\(\cos(\theta) = \cos(\theta + 2k\pi)\)[/tex] for any integer [tex]\(k\)[/tex]. Thus, we can reduce [tex]\(3\pi\)[/tex] by subtracting [tex]\(2\pi\)[/tex] until the angle lies within the standard range of [tex]\([0, 2\pi)\)[/tex].
[tex]\[
3\pi - 2\pi = \pi
\][/tex]
3. Finding [tex]\(\cos(\pi)\)[/tex]: Now that we have reduced the angle to [tex]\(\pi\)[/tex],
[tex]\[
\cos(\pi) = -1
\][/tex]
Thus, the value of [tex]\(\cos(3\pi)\)[/tex] is [tex]\(-1\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{-1}
\][/tex]
