Question

evaluate tan(4π/3). write an exact answer in simplest form.

Explanation:

Step1: Find the reference angle

The angle \(\frac{4\pi}{3}\) is in the third quadrant. The reference angle \(\theta'\) is \(\frac{4\pi}{3}-\pi=\frac{\pi}{3}\).

Step2: Determine the sign of the tangent function

In the third quadrant, \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and both \(\sin\theta\) and \(\cos\theta\) are negative. So \(\tan\theta\) is positive.

Step3: Evaluate using the reference angle

We know that \(\tan\frac{\pi}{3}=\sqrt{3}\). Since \(\tan\frac{4\pi}{3}\) has the same value as \(\tan\frac{\pi}{3}\) (because of the sign analysis in Step 2), \(\tan\frac{4\pi}{3}=\sqrt{3}\).

Answer:

\(\sqrt{3}\)