The value of $\tan\left(\frac{5\pi}{3}\right)$ can be found using the properties of the tangent function and the unit circle.

Step 1: Determine the reference angle

The angle $\frac{5\pi}{3}$ is in the fourth quadrant because $5\pi/3$ radians is equivalent to $300^\circ$. The reference angle is calculated as: $360^\circ - 300^\circ = 60^\circ$ Or in radians: $2\pi - \frac{5\pi}{3} = \frac{\pi}{3}$

Step 2: Find the tangent of the reference angle

The reference angle is $\frac{\pi}{3}$, and we know: $\tan\left(\frac{\pi}{3}\right) = \sqrt{3}$

Step 3: Determine the sign of the tangent

Since the angle $\frac{5\pi}{3}$ is in the fourth quadrant, and tangent is negative in the fourth quadrant: $\tan\left(\frac{5\pi}{3}\right) = -\sqrt{3}$

So, the value of $\tan\left(\frac{5\pi}{3}\right)$ is $-\sqrt{3}$.

Would you like any further details or explanations on this? Here are five related questions you might find interesting:

1. What is the value of $\tan\left(\frac{7\pi}{6}\right)$?
2. How do you find the reference angle for any given radian measure?
3. What is the relationship between tangent and cotangent functions?
4. How does the unit circle help in determining the sign of trigonometric functions?
5. Can you explain how to derive the value of $\tan\left(\frac{\pi}{4}\right)$?

Tip: Remember that the sign of the tangent function depends on the quadrant in which the angle lies.