To find the value of [tex]\(\tan 30^\circ\)[/tex], we need to recall that [tex]\(\tan \theta\)[/tex] is defined as the ratio of the opposite side to the adjacent side in a right triangle:
[tex]\[\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\][/tex]
For a 30-degree angle, we can use the properties of an equilateral triangle. An equilateral triangle has all sides of equal length and all angles equal to 60 degrees. If we cut an equilateral triangle in half, we can create a right triangle with angles of 30 degrees, 60 degrees, and 90 degrees, and the hypotenuse being the original side of the equilateral triangle.
Let's assume the side length of the equilateral triangle is 2 units. Then, when we cut it in half:
- The hypotenuse (original side of the equilateral triangle) will still be 2 units.
- The side opposite the 30-degree angle will be 1 unit (half of the original side).
- The adjacent side can be found using the Pythagorean theorem:
[tex]\[
\text{adjacent}^2 + 1^2 = 2^2
\][/tex]
[tex]\[
\text{adjacent}^2 + 1 = 4
\][/tex]
[tex]\[
\text{adjacent}^2 = 3
\][/tex]
[tex]\[
\text{adjacent} = \sqrt{3}
\][/tex]
Thus, in our right triangle, for a 30-degree angle, we have:
- Opposite side = 1 unit
- Adjacent side = [tex]\(\sqrt{3}\)[/tex] units
So, the tangent of 30 degrees is:
[tex]\[
\tan 30^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{\sqrt{3}}
\][/tex]
This simplifies approximately to:
[tex]\[
\tan 30^\circ \approx 0.5773502691896257
\][/tex]
Comparing this with the given options:
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. 1
F. [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
The correct answer is [tex]\(\frac{1}{\sqrt{3}}\)[/tex], which matches option F.
So, the value of [tex]\(\tan 30^\circ\)[/tex] is:
[tex]\[ \boxed{\frac{1}{\sqrt{3}}} \][/tex]
