To find the value of [tex]\(\tan 60^\circ\)[/tex], we start by understanding the definition of the tangent function in trigonometry. The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the adjacent side.
Mathematically,
[tex]\[
\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}
\][/tex]
For the specific angle of [tex]\(60^\circ\)[/tex], we commonly use the properties of a 30°-60°-90° triangle. In a 30°-60°-90° triangle, the sides are in the ratio:
[tex]\[
1 : \sqrt{3} : 2
\][/tex]
where 1 is the side opposite 30°, [tex]\(\sqrt{3}\)[/tex] is the side opposite 60°, and 2 is the hypotenuse.
Given this ratio, for [tex]\(\tan 60^\circ\)[/tex]:
[tex]\[
\tan 60^\circ = \frac{\text{Opposite to } 60^\circ}{\text{Adjacent to } 60^\circ} = \frac{\sqrt{3}}{1} = \sqrt{3}
\][/tex]
Thus, the value of [tex]\(\tan 60^\circ\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
In decimal form, [tex]\(\sqrt{3}\)[/tex] approximately equals:
[tex]\[
1.7320508075688772
\][/tex]
Therefore, [tex]\(\tan 60^\circ = 1.7320508075688772\)[/tex].
This understanding and calculation show that the exact value of [tex]\(\tan 60^\circ\)[/tex] is [tex]\(\sqrt{3}\)[/tex], which in decimal form is approximately [tex]\(1.7320508075688772\)[/tex].
