To solve for [tex]\(\tan 30^\circ\)[/tex], we need to recall a fundamental property of trigonometric functions in a 30-60-90 triangle. In such a right triangle, the angles are [tex]\(30^\circ\)[/tex], [tex]\(60^\circ\)[/tex], and [tex]\(90^\circ\)[/tex]. The sides of a 30-60-90 triangle have a characteristic ratio:
- The side opposite the 30-degree angle is [tex]\( \frac{1}{2} \)[/tex] of the hypotenuse.
- The side opposite the 60-degree angle is [tex]\( \frac{\sqrt{3}}{2} \)[/tex] of the hypotenuse.
- The hypotenuse is the longest side.
For a 30-60-90 triangle where the hypotenuse is taken as 1 (unit circle representation), the lengths of the sides are:
- Opposite the 30-degree angle: [tex]\(\frac{1}{2}\)[/tex]
- Opposite the 60-degree angle: [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
By definition, the tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Therefore,
[tex]\[
\tan 30^\circ = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}
\][/tex]
Simplify this ratio:
[tex]\[
\tan 30^\circ = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}
\][/tex]
So, [tex]\(\tan 30^\circ\)[/tex] is [tex]\(\frac{1}{\sqrt{3}}\)[/tex].
Comparing this with the given options, we see that the correct answer is:
E. [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
From evaluating this answer numerically, we find [tex]\(\tan 30^\circ ≈ 0.5773502691896258\)[/tex], which confirms the ratio found in choice E is correct. Therefore, the answer is [tex]\( \boxed{E} \)[/tex].
