Question

question given the following unit circle, rotate green dot to the appropriate angle and then find the exact value of the function. \\( \tan 240^\circ \\)

Explanation:

Step1: Find the reference angle

The angle $240^\circ$ is in the third quadrant. The reference angle is calculated as $240^\circ - 180^\circ = 60^\circ$.

Step2: Determine the sign of $\tan$

In the third quadrant, both sine and cosine are negative, and $\tan\theta=\frac{\sin\theta}{\cos\theta}$, so $\tan$ is positive in the third quadrant.

Step3: Calculate $\tan$ of the reference angle

We know that $\tan60^\circ=\sqrt{3}$. Since the sign of $\tan240^\circ$ is positive (from Step 2) and the reference angle is $60^\circ$, we have $\tan240^\circ = \tan(180^\circ + 60^\circ)=\tan60^\circ=\sqrt{3}$.

Answer:

$\sqrt{3}$