To determine the value of [tex]\(\tan 60^\circ\)[/tex], we can use our knowledge of trigonometric values for common angles.
1. Recall that the tangent function ([tex]\(\tan\)[/tex]) for an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
2. For a [tex]\(60^\circ\)[/tex] angle, we can use the properties of a 30-60-90 triangle. In such a triangle, the ratio of the sides opposite the [tex]\(30^\circ\)[/tex], [tex]\(60^\circ\)[/tex], and [tex]\(90^\circ\)[/tex] angles are [tex]\(1 : \sqrt{3} : 2\)[/tex], respectively.
3. For the [tex]\(60^\circ\)[/tex] angle, the side opposite to it is [tex]\(\sqrt{3}\)[/tex] and the adjacent side is [tex]\(1\)[/tex]. Therefore,
[tex]\[
\tan 60^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{3}}{1} = \sqrt{3}
\][/tex]
Thus, the value of [tex]\(\tan 60^\circ\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
Given the options:
A. [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
B. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
C. [tex]\(\frac{1}{2}\)[/tex]
D. [tex]\(\sqrt{3}\)[/tex]
E. 1
F. [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
The correct answer is:
[tex]\[ \boxed{\sqrt{3}} \][/tex]
