To determine the value of [tex]\(\tan(60^\circ)\)[/tex], let's start by recalling some fundamental trigonometric properties and values.
First, recall that the tangent of an angle in trigonometry is defined as the ratio of the sine of the angle to the cosine of the angle:
[tex]\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\][/tex]
For the specific angle of [tex]\(60^\circ\)[/tex], we use the exact values for the sine and cosine of [tex]\(60^\circ\)[/tex]:
[tex]\[
\sin(60^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
\cos(60^\circ) = \frac{1}{2}
\][/tex]
Now, substituting these values into the tangent formula, we have:
[tex]\[
\tan(60^\circ) = \frac{\sin(60^\circ)}{\cos(60^\circ)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}
\][/tex]
To simplify this fraction, we can multiply the numerator and the denominator by 2:
[tex]\[
\tan(60^\circ) = \frac{\frac{\sqrt{3}}{2} \times 2}{\frac{1}{2} \times 2} = \frac{\sqrt{3}}{1} = \sqrt{3}
\][/tex]
Thus, the value of [tex]\(\tan(60^\circ)\)[/tex] is:
[tex]\[
\sqrt{3} \approx 1.7320508075688772
\][/tex]
Out of the given options, the correct answer is:
[tex]\[
\sqrt{3}
\][/tex]
