To determine the value of [tex]\(\tan 60^\circ\)[/tex], let's analyze this trigonometric function for the given angle.
We know from trigonometry that tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side, i.e.,
[tex]\[
\tan \theta = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
For the angle [tex]\(60^\circ\)[/tex], which is a standard angle, its tangent value is commonly derived from the properties of a 30-60-90 triangle. In a 30-60-90 triangle, the sides are in the ratio of:
[tex]\[
1 : \sqrt{3} : 2
\][/tex]
Specifically:
- The side opposite the 30° angle is [tex]\(1\)[/tex].
- The side opposite the 60° angle is [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse is [tex]\(2\)[/tex].
Thus, for [tex]\(\tan 60^\circ\)[/tex], where the angle is 60°:
[tex]\[
\tan 60^\circ = \frac{\text{opposite to 60°}}{\text{adjacent to 60°}} = \frac{\sqrt{3}}{1} = \sqrt{3}
\][/tex]
We now compare this value with the options given:
A. [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
B. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
C. [tex]\(\frac{1}{2}\)[/tex]
D. 1
E. [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
F. [tex]\(\sqrt{3}\)[/tex]
The correct choice is [tex]\(\sqrt{3}\)[/tex], which corresponds to:
F. [tex]\(\sqrt{3}\)[/tex]
So, the value of [tex]\(\tan 60^\circ\)[/tex] is [tex]\(\sqrt{3}\)[/tex]. Therefore, the correct answer is:
[tex]\[
\boxed{F}
\][/tex]
