The value of \( \tan 60^\circ \) can be derived from the properties of an equilateral triangle or using the unit circle.
In an equilateral triangle, each angle is \( 60^\circ \). If we consider such a triangle with a side length of 1, we can drop a perpendicular from one vertex to the midpoint of the opposite side, creating two 30-60-90 triangles. In these triangles:
- The side opposite the \( 60^\circ \) angle is \( \frac{\sqrt{3}}{2} \).
- The side opposite the \( 30^\circ \) angle is \( \frac{1}{2} \).
The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side:
\[ \tan 60^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}. \]
Thus, the value of \( \tan 60^\circ \) is \( \sqrt{3} \).
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