To determine the value of [tex]\(\tan 60^{\circ}\)[/tex], we start by understanding the trigonometric tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle.
Specifically, for an angle of [tex]\(60^{\circ}\)[/tex], we can use geometry or trigonometric identities:
1. Consider the equilateral triangle where each angle is [tex]\(60^\circ\)[/tex]. By splitting this triangle into two right-angled triangles, you create two [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangles.
2. In a [tex]\(30^\circ-60^\circ-90^\circ\)[/tex] triangle:
- The side opposite the [tex]\(30^\circ\)[/tex] angle is half the hypotenuse.
- The side opposite the [tex]\(60^\circ\)[/tex] angle is [tex]\(\sqrt{3}/2\)[/tex] times the hypotenuse.
- If we let the hypotenuse be 1, then the side opposite [tex]\(30^\circ\)[/tex] is [tex]\(1/2\)[/tex], and the side opposite [tex]\(60^\circ\)[/tex] is [tex]\(\sqrt{3}/2\)[/tex].
3. The tangent of an angle in a right triangle is given by the ratio of the length of the opposite side to the length of the adjacent side.
- Therefore, [tex]\(\tan 60^{\circ}\)[/tex] is the length of the side opposite [tex]\(60^\circ\)[/tex] divided by the length of the side adjacent to [tex]\(60^\circ\)[/tex].
4. In this triangle, the side opposite [tex]\(60^\circ\)[/tex] is [tex]\(\sqrt{3}/2\)[/tex], and the side adjacent to [tex]\(60^\circ\)[/tex] is [tex]\(1/2\)[/tex]. Therefore, we have:
[tex]\[
\tan 60^{\circ} = \frac{\left(\sqrt{3}/2\right)}{\left(1/2\right)} = \sqrt{3}
\][/tex]
Thus, the value of [tex]\(\tan 60^{\circ}\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
Therefore, the correct answer is:
B. [tex]\(\sqrt{3}\)[/tex]
