To determine the value of [tex]\(\cos 30^{\circ}\)[/tex], let's analyze the options step-by-step.
1. Identify the reference angle visually:
The angle [tex]\(30^\circ\)[/tex] is a common angle in trigonometry, often associated with the 30-60-90 right triangle. In such a triangle, the sides have a specific ratio:
- The length of the side opposite the [tex]\(30^\circ\)[/tex] angle is [tex]\(\frac{1}{2}\)[/tex].
- The length of the side opposite the [tex]\(60^\circ\)[/tex] angle is [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
- The hypotenuse has a length of 1.
2. Calculate [tex]\(\cos 30^\circ\)[/tex]:
The cosine of an angle in a right triangle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. For a [tex]\(30^\circ\)[/tex] angle:
- Adjacent side (to [tex]\(30^\circ\)[/tex]) = [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Hypotenuse = 1
Therefore,
[tex]\[
\cos 30^\circ = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{\frac{\sqrt{3}}{2}}{1} = \frac{\sqrt{3}}{2}
\][/tex]
3. Match this result to the given multiple-choice answers:
- Option A: [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
- Option B: 1
- Option C: [tex]\(\sqrt{3}\)[/tex]
- Option D: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- Option E: [tex]\(\frac{1}{2}\)[/tex]
- Option F: [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
Among these options, the correct one that matches our result [tex]\(\frac{\sqrt{3}}{2}\)[/tex] is:
D. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
In conclusion, the value of [tex]\(\cos 30^{\circ}\)[/tex] is [tex]\(\frac{\sqrt{3}}{2}\)[/tex], which corresponds to option D.
