To find the exact value of [tex]\(\cos 2\pi\)[/tex], let's start by understanding the basic properties of the cosine function and its behavior on the unit circle.
The cosine function is periodic with a period of [tex]\(2\pi\)[/tex]. This means that the values of cosine repeat every [tex]\(2\pi\)[/tex] radians. Specifically, [tex]\(\cos(\theta + 2\pi k) = \cos(\theta)\)[/tex] for any integer [tex]\(k\)[/tex].
The unit circle is a circle with radius 1 centered at the origin of the coordinate system. In trigonometry, the angle is measured from the positive x-axis counterclockwise.
1. Angle of 0 radians (0 degrees): At 0 radians, the point on the unit circle is [tex]\((1, 0)\)[/tex]. The cosine of 0 radians is the x-coordinate, which is 1.
[tex]\[
\cos 0 = 1
\][/tex]
2. Angle of [tex]\(2\pi\)[/tex] radians (360 degrees): If you start at 0 radians and travel one full revolution counterclockwise, you end up back at the starting point, which is also [tex]\((1, 0)\)[/tex]. Hence, the cosine of [tex]\(2\pi\)[/tex] radians is the x-coordinate of this point.
[tex]\[
\cos 2\pi = 1
\][/tex]
Therefore, the exact value of [tex]\(\cos 2\pi\)[/tex] is [tex]\(\boxed{1}\)[/tex].
