To determine the value of [tex]\(\tan(60^\circ)\)[/tex], we start by considering the properties of trigonometric functions and special angles. The angle [tex]\(60^\circ\)[/tex] is a well-known angle in trigonometry, and it often appears in the context of equilateral and 30-60-90 triangles.
1. Special Triangles:
- Consider a 30-60-90 triangle. In this type of triangle, the ratios of the sides are well known:
- The side opposite the [tex]\(30^\circ\)[/tex] angle is [tex]\(1\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle is [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse is [tex]\(2\)[/tex].
2. Definition of Tangent:
- The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. For [tex]\(\tan(60^\circ)\)[/tex], the side opposite the [tex]\(60^\circ\)[/tex] angle is [tex]\(\sqrt{3}\)[/tex] and the side adjacent to it (which is the side opposite the [tex]\(30^\circ\)[/tex] angle) is [tex]\(1\)[/tex].
3. Calculating:
[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].
Among the given options:
- [tex]\(\frac{1}{2}\)[/tex]
- [tex]\(\sqrt{3}\)[/tex]
- [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
The correct answer is [tex]\(\sqrt{3}\)[/tex].
