To determine the value of [tex]\(\tan 60^{\circ}\)[/tex], we can use the properties of a 30-60-90 triangle. In such a triangle, the ratios of the sides are well known.
A 30-60-90 triangle has its sides in the ratio:
- Opposite the 30° angle is [tex]\(a\)[/tex]
- Opposite the 60° angle is [tex]\(a \sqrt{3}\)[/tex]
- The hypotenuse is [tex]\(2a\)[/tex]
For [tex]\(\tan \theta\)[/tex], we use the definition of tangent in a right triangle:
[tex]\[
\tan \theta = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Using these properties for the angle [tex]\(60^{\circ}\)[/tex]:
- The side opposite [tex]\(60^{\circ}\)[/tex] is [tex]\(a\sqrt{3}\)[/tex]
- The side adjacent to [tex]\(60^{\circ}\)[/tex] is [tex]\(a\)[/tex]
Thus,
[tex]\[
\tan 60^{\circ} = \frac{a \sqrt{3}}{a} = \sqrt{3}
\][/tex]
Therefore, the value of [tex]\(\tan 60^{\circ}\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
So, the correct answer is:
E. [tex]\(\sqrt{3}\)[/tex]
