To determine the value of [tex]\(\tan\left(60^\circ\right)\)[/tex], we start by recalling that the tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side. However, for this specific problem, we'll consider the standard trigonometric values for specific angles.
### Step-by-Step Solution:
1. Convert the Angle to Radians:
- Angles can be measured in degrees or radians. For many trigonometric functions, including tangent, it's common to convert the angle from degrees to radians. The conversion from degrees to radians is done using the formula:
[tex]\[
\text{Radians} = \text{Degrees} \times \frac{\pi}{180}
\][/tex]
- For [tex]\(60^\circ\)[/tex]:
[tex]\[
60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}
\][/tex]
2. Recall the Tangent Value for [tex]\(\frac{\pi}{3}\)[/tex]:
- Standard trigonometric values are often memorized or found in reference tables. For an angle of [tex]\(\frac{\pi}{3}\)[/tex] radians (or [tex]\(60^\circ\)[/tex]), the value of the tangent function is:
[tex]\[
\tan\left(\frac{\pi}{3}\right) = \sqrt{3}
\][/tex]
Thus, the value of [tex]\(\tan(60^\circ)\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
Comparing this with the provided choices:
- [tex]\(\frac{1}{2}\)[/tex]
- [tex]\(\sqrt{3}\)[/tex]
- [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
- [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
The correct answer is:
[tex]\[
\boxed{\sqrt{3}}
\][/tex]
