To determine the value of [tex]\(\tan(60^\circ)\)[/tex], let's start by understanding the relationship between angles and their tangent values.
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. Specifically, for [tex]\(\tan(60^\circ)\)[/tex]:
[tex]\[
\tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
In a 30-60-90 right triangle, the ratios of the sides are known:
- The side opposite the 30° angle is [tex]\(a\)[/tex],
- The side opposite the 60° angle is [tex]\(a\sqrt{3}\)[/tex],
- The hypotenuse is [tex]\(2a\)[/tex].
This results in:
[tex]\[
\tan(60^\circ) = \frac{\text{opposite side of } 60^\circ}{\text{adjacent side of } 60^\circ} = \frac{a\sqrt{3}}{a} = \sqrt{3}
\][/tex]
Hence, the value of [tex]\(\tan(60^\circ)\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
The correct answer is:
[tex]\(\sqrt{3}\)[/tex]
