To find the value of [tex]\(\tan(60^\circ)\)[/tex], we need to use trigonometric identities and properties.
Firstly, recall the definition of the tangent function in terms of a right-angled triangle:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
For the specific angle of [tex]\(60^\circ\)[/tex], we use a special right triangle, the 30-60-90 triangle, where the side lengths are in the ratio [tex]\(1: \sqrt{3}: 2\)[/tex]. The sides opposite the [tex]\(30^\circ\)[/tex], [tex]\(60^\circ\)[/tex], and [tex]\(90^\circ\)[/tex] angles have lengths, respectively, of [tex]\(1\)[/tex], [tex]\(\sqrt{3}\)[/tex], and [tex]\(2\)[/tex].
Considering the angle [tex]\(60^\circ\)[/tex]:
- The side opposite [tex]\(60^\circ\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
- The side adjacent to [tex]\(60^\circ\)[/tex] is [tex]\(1\)[/tex].
Thus, we have:
[tex]\[
\tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{3}}{1} = \sqrt{3}
\][/tex]
Therefore, the value of [tex]\(\tan(60^\circ)\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
The correct choice from the given options is:
[tex]\[
\sqrt{3}
\][/tex]
