Evaluate. Write your answer in simplified, rationalized form. Do not round.tan60°=

1. Break Down the Problem

To evaluate $\tan 60^\circ$, we need to understand the definition of the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.

2. Relevant Concepts

The tangent function can also be derived from the unit circle or trigonometric identities. For a special triangle, specifically the 30-60-90 triangle:

• The lengths of the sides are in the ratio $1 : \sqrt{3} : 2$.
• Therefore, for $\tan 60^\circ$:

$\tan 60^\circ = \frac{\text{opposite}}{\text{adjacent}}$

3. Analysis and Detail

In a 30-60-90 triangle, where:

• The side opposite the $60^\circ$ angle has a length of $\sqrt{3}$,
• The side adjacent to the $60^\circ$ angle has a length of $1$.

Thus:

$\tan 60^\circ = \frac{\sqrt{3}}{1} = \sqrt{3}$

4. Verify and Summarize

We can verify the calculations by checking the properties of the triangle. The calculations confirm the known property of $\tan$ at $60^\circ$.

Final Answer

$\tan 60^\circ = \sqrt{3}$