To determine the value of [tex]\(\tan 30^\circ\)[/tex], we can consider the exact values of trigonometric functions for specific angles, often derived from the properties of special triangles, such as the 30-60-90 triangle.
In a 30-60-90 triangle, the angles are [tex]\(30^\circ\)[/tex], [tex]\(60^\circ\)[/tex], and [tex]\(90^\circ\)[/tex]. The side lengths in such a triangle are in the ratio [tex]\(1:\sqrt{3}:2\)[/tex], where:
- The side opposite the [tex]\(30^\circ\)[/tex] angle (shorter leg) is [tex]\(1\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle (longer leg) is [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse (opposite the [tex]\(90^\circ\)[/tex] angle) is [tex]\(2\)[/tex].
For [tex]\(\tan 30^\circ\)[/tex]:
[tex]\[
\tan 30^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{\sqrt{3}}
\][/tex]
Thus, the exact value of [tex]\(\tan 30^\circ\)[/tex] is [tex]\(\frac{1}{\sqrt{3}}\)[/tex].
To ensure the accuracy of this value, we approximate the numerical value of [tex]\(\frac{1}{\sqrt{3}}\)[/tex]:
[tex]\[
\frac{1}{\sqrt{3}} \approx 0.5773502691896257
\][/tex]
Given the multiple-choice options, the correct representation is:
F. [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
