To determine the value of [tex]\(\tan(60^\circ)\)[/tex], we first need to understand that 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 with respect to that angle.
1. Convert the angle to radians:
Angles can be measured in degrees or radians. To proceed with trigonometric calculations, it's often convenient to convert the angle from degrees to radians.
The conversion formula is:
[tex]\[
\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}
\][/tex]
For [tex]\(60^\circ\)[/tex]:
[tex]\[
60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}
\][/tex]
2. Use the tangent function:
Now we need to find the value of [tex]\(\tan\left(\frac{\pi}{3}\right)\)[/tex]. The value of the tangent function for the angle [tex]\(\frac{\pi}{3}\)[/tex] (or [tex]\(60^\circ\)[/tex]) is a known trigonometric fact.
3. Result:
The value of [tex]\(\tan\left(\frac{\pi}{3}\right)\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
Therefore, the value of [tex]\(\tan(60^\circ)\)[/tex] is:
[tex]\[
\sqrt{3}
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{\sqrt{3}}
\][/tex]
