To determine the value of [tex]\(\tan(60^\circ)\)[/tex], we need to understand the trigonometric properties of special angles. The tangent function, [tex]\(\tan(\theta)\)[/tex], is defined as the ratio of the sine and cosine functions of that angle:
[tex]\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\][/tex]
For the specific angle [tex]\(\theta = 60^\circ\)[/tex]:
1. The sine of 60 degrees is known to be [tex]\(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)[/tex].
2. The cosine of 60 degrees is known to be [tex]\(\cos(60^\circ) = \frac{1}{2}\)[/tex].
Using these values, we can calculate the tangent of 60 degrees:
[tex]\[
\tan(60^\circ) = \frac{\sin(60^\circ)}{\cos(60^\circ)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{2} \times \frac{2}{1} = \sqrt{3}
\][/tex]
Therefore, the value of [tex]\(\tan(60^\circ)\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
Given the answer 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]\(\sqrt{3}\)[/tex].
