To determine the value of [tex]\(\tan 30^\circ\)[/tex], we can begin by recalling that the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
For a 30-degree angle, we often use the properties of a special right triangle, the 30-60-90 triangle. The sides of such a triangle are in the ratio 1 : [tex]\(\sqrt{3}\)[/tex] : 2. Specifically:
- The side opposite the 30-degree angle is [tex]\(1\)[/tex].
- The side opposite the 60-degree angle is [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse is [tex]\(2\)[/tex].
Given this information, the formula for the tangent of a 30-degree angle is:
[tex]\[
\tan 30^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{\sqrt{3}}
\][/tex]
To select the proper answer from the provided options, we look for the choice that matches our calculated value. The possible answers provided are:
A. 1
B. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
C. [tex]\(\sqrt{3}\)[/tex]
D. [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
E. [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
F. [tex]\(\sqrt{2}\)[/tex]
After examining these options, we see that:
- Option D is [tex]\(\frac{1}{\sqrt{3}}\)[/tex], which matches our calculated value.
Therefore, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
