To find the value of [tex]\(\tan(60^{\circ})\)[/tex], let's go through the steps to understand why our answer is [tex]\(\sqrt{3}\)[/tex].
1. Convert Degrees to Radians:
Since trigonometric functions in standard mathematical contexts usually expect angles in radians, we convert [tex]\(60^{\circ}\)[/tex] to radians. We use the fact that [tex]\(180^{\circ} = \pi\)[/tex] radians.
[tex]\[
60^{\circ} = \frac{60 \cdot \pi}{180} = \frac{\pi}{3} \text{ radians}
\][/tex]
2. Understanding 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. For trigonometric functions at standard angles, we can use known values or unit circle definitions.
3. Special Angle Values:
The value of [tex]\(\tan\left(\frac{\pi}{3}\right)\)[/tex] is one of the standard trigonometric values, which can be derived from the 30-60-90 special triangle or directly from the unit circle.
[tex]\[
\tan\left(60^{\circ}\right) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3}
\][/tex]
Therefore, the value of [tex]\(\tan(60^{\circ})\)[/tex] is:
[tex]\[
\sqrt{3}
\][/tex]
So, from the given options, the correct answer is:
[tex]\[
\boxed{\sqrt{3}}
\][/tex]
