To find the value of [tex]\(\tan(60^\circ)\)[/tex], follow these steps:
1. Understand the definition of the tangent function:
The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
2. Recall the properties of a 30-60-90 triangle:
A 30-60-90 triangle is a special right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees.
- The side lengths of a 30-60-90 triangle are in the ratio [tex]\(1 : \sqrt{3} : 2\)[/tex].
- For the 60-degree angle, the side opposite to it is [tex]\(\sqrt{3}\)[/tex] and the side adjacent to it is [tex]\(1\)[/tex].
3. Apply the tangent function:
For [tex]\(\tan(60^\circ)\)[/tex]:
[tex]\[
\tan(60^\circ) = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\sqrt{3}}{1} = \sqrt{3}
\][/tex]
4. Compare with the given options:
After calculating, we find that [tex]\(\tan(60^\circ) = \sqrt{3}\)[/tex].
Thus, the value of [tex]\(\tan(60^\circ)\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
The correct answer is [tex]\(\sqrt{3}\)[/tex].
