To find the value of [tex]\(\tan(60^\circ)\)[/tex], follow these steps:
1. Identify the Angle: We are dealing with an angle of [tex]\(60^\circ\)[/tex].
2. Recall the Definition of Tangent: 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. For the angle [tex]\(60^\circ\)[/tex], which is one of the special angles in trigonometry, this ratio is well-known.
3. Use the Known Values: For the special angles, particularly in a 30-60-90 triangle, the sides have a specific ratio:
- The side opposite the [tex]\(30^\circ\)[/tex] angle has a length of [tex]\(1\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle (which we need) has a length of [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse has a length of [tex]\(2\)[/tex].
4. Calculate Tangent: Using the definition of tangent:
[tex]\[
\tan(60^\circ) = \frac{\text{length of the opposite side}}{\text{length of the adjacent side}} = \frac{\sqrt{3}}{1} = \sqrt{3}
\][/tex]
5. Confirm by comparing with standard trigonometric values:
- We know that [tex]\(\tan(60^\circ) = \sqrt{3}\)[/tex].
Therefore, the value of [tex]\(\tan(60^\circ)\)[/tex] is [tex]\(\sqrt{3}\)[/tex]. Hence, the correct answer is:
[tex]\[
\boxed{\sqrt{3}}
\][/tex]
