To determine [tex]\(\cos 30^\circ\)[/tex], we need to use the properties and side lengths of a 30-60-90 triangle.
In a 30-60-90 triangle, the side lengths are in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex]. Specifically:
- The side opposite the 30° angle (the shortest side) is [tex]\(1\)[/tex].
- The side opposite the 60° angle (the longer leg) is [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse (the side opposite the 90° angle) is [tex]\(2\)[/tex].
The cosine of an angle in a right triangle is defined as the length of the adjacent side divided by the length of the hypotenuse.
For [tex]\(\cos 30^\circ\)[/tex]:
- The adjacent side to the 30° angle is the side opposite the 60° angle, which has a length of [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse is the side opposite the 90° angle, which has a length of [tex]\(2\)[/tex].
Therefore, we calculate [tex]\(\cos 30^\circ\)[/tex] as follows:
[tex]\[
\cos 30^\circ = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}
\][/tex]
Thus, the correct answer is:
D. [tex]\(\frac{\sqrt{3}}{2}\)[/tex], which numerically is approximately 0.8660254037844386.
