To determine the value of [tex]\(\cos 30^\circ\)[/tex], let's go through the steps thoroughly.
1. Recall that in trigonometry, the cosine of an angle in a right triangle is defined as the ratio of the adjacent side over the hypotenuse.
2. For a [tex]\(30^\circ\)[/tex] angle in an equilateral triangle cut in half, we can use the special triangles properties:
- An equilateral triangle has all sides equal and all angles equal to [tex]\(60^\circ\)[/tex].
- Cutting the triangle in half gives us a right triangle with angles [tex]\(30^\circ\)[/tex], [tex]\(60^\circ\)[/tex], and [tex]\(90^\circ\)[/tex].
- The sides of this right triangle (half of the equilateral triangle) have ratios of 1 (opposite the [tex]\(30^\circ\)[/tex]), [tex]\(\sqrt{3}\)[/tex] (opposite the [tex]\(60^\circ\)[/tex]), and 2 (the hypotenuse).
3. For the [tex]\(30^\circ\)[/tex] triangle:
- The ratio of the adjacent side (which is [tex]\(\sqrt{3}\)[/tex]) to the hypotenuse [tex]\(2\)[/tex] in a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle is thus:
[tex]\[
\cos 30^\circ = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{\sqrt{3}}{2}
\][/tex]
Given this information, let's check the provided options:
- A. [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
- B. 1
- C. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- D. [tex]\(\sqrt{3}\)[/tex]
- E. [tex]\(\frac{1}{2}\)[/tex]
- F. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
From our reconfirmed calculation, the correct value of [tex]\(\cos 30^\circ\)[/tex] matches option C.
Therefore, the correct answer is C. [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
