What is [tex]\tan 30^{\circ}[/tex]?

A. [tex]\frac{2}{\sqrt{3}}[/tex]

B. [tex]\frac{\sqrt{3}}{2}[/tex]

C. [tex]\sqrt{2}[/tex]

D. [tex]\sqrt{3}[/tex]

E. [tex]\frac{1}{\sqrt{3}}[/tex]

F. 1



Answer :

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].