Find the exact value of tangent, 240, degreestan240 ∘ in simplest form with a rational denominator

1. Break Down the Problem

We need to find the exact value of $\tan(240^\circ)$.

2. Relevant Concepts

The angle $240^\circ$ is located in the third quadrant. The tangent function can be calculated using the reference angle, which in this case is given by: $\text{Reference angle} = 240^\circ - 180^\circ = 60^\circ$ In the third quadrant, the tangent function is positive.

The formula for tangent is: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ For the reference angle $60^\circ$: $\tan(60^\circ) = \sqrt{3}$

3. Analysis and Detail

Since $240^\circ$ is in the third quadrant, where tangent is positive, we have: $\tan(240^\circ) = \tan(60^\circ) = \sqrt{3}$

4. Verify and Summarize

The value of $\tan(240^\circ)$ has been verified through its reference angle and which quadrant it is located in.

Final Answer

$\tan(240^\circ) = \sqrt{3}$