To find the value of [tex]\(\tan 60^{\circ}\)[/tex], let's follow the steps necessary to determine it:
1. Recall 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. Consider the special right triangles: Specifically, consider the [tex]\(30^{\circ}-60^{\circ}-90^{\circ}\)[/tex] triangle. In this type of triangle, the ratios of the sides have known values:
- The side opposite the [tex]\(30^{\circ}\)[/tex] angle is the shortest, which is [tex]\(1\)[/tex].
- The side opposite the [tex]\(60^{\circ}\)[/tex] angle is [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse (opposite the [tex]\(90^{\circ}\)[/tex] angle) is [tex]\(2\)[/tex].
3. Calculate [tex]\(\tan 60^{\circ}\)[/tex]: Using this information, the tangent of [tex]\(60^{\circ}\)[/tex] is the ratio of the length of the side opposite the [tex]\(60^{\circ}\)[/tex] angle ([tex]\(\sqrt{3}\)[/tex]) to the length of the side adjacent to the [tex]\(60^{\circ}\)[/tex] angle (which is [tex]\(1\)[/tex]):
[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].
4. Match with the given choices:
- A. [tex]\(\frac{1}{2}\)[/tex]
- B. [tex]\(\sqrt{3}\)[/tex]
- C. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
The correct answer is:
- B. [tex]\(\sqrt{3}\)[/tex]
Thus, [tex]\(\tan 60^{\circ} = \sqrt{3}\)[/tex] (option B).
