To find [tex]\(\cos 60^\circ\)[/tex], let's go through the steps and reasoning required.
1. Understand the Unit Circle and Cosine Function:
The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. When considering the unit circle, where the radius (hypotenuse) is always 1, the cosine of an angle is simply the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
2. Special Angles in Trigonometry:
The angle [tex]\(60^\circ\)[/tex] is one of the standard angles that have well-known sine and cosine values. These standard angles are typically [tex]\(0^\circ\)[/tex], [tex]\(30^\circ\)[/tex], [tex]\(45^\circ\)[/tex], [tex]\(60^\circ\)[/tex], and [tex]\(90^\circ\)[/tex].
3. Cosine Value of [tex]\(60^\circ\)[/tex]:
The cosine of [tex]\(60^\circ\)[/tex] is a known value that can be found on the unit circle. Specifically:
[tex]\[
\cos 60^\circ = \frac{1}{2}
\][/tex]
4. Matching with the Choices Provided:
- Option A: [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
- Option B: [tex]\(\sqrt{3}\)[/tex]
- Option C: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Option D: [tex]\(\frac{1}{2}\)[/tex]
- Option E: 1
- Option F: [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
From these choices, the correct match for [tex]\(\cos 60^\circ\)[/tex] is:
[tex]\[
\boxed{\frac{1}{2}}
\][/tex]
