To find the value of $\tan\left(\frac{13\pi}{6}\right)$, we can first convert this angle to a reference angle within the primary interval $[0, 2\pi)$.

Step 1: Convert to an equivalent angle in the range $[0, 2\pi)$

$\frac{13\pi}{6} = 2\pi + \frac{\pi}{6}$

Since $2\pi$ represents a full circle, we can subtract $2\pi$ to find the reference angle:

$\frac{13\pi}{6} - 2\pi = \frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6}$

Step 2: Determine the tangent of the reference angle

Now, we know that $\frac{13\pi}{6}$ has the same tangent value as $\frac{\pi}{6}$ because $\tan(\theta + 2\pi) = \tan(\theta)$.

$\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}$

Conclusion

$\tan\left(\frac{13\pi}{6}\right) = \frac{1}{\sqrt{3}}$

Or, in a simplified form:

$\tan\left(\frac{13\pi}{6}\right) = \frac{\sqrt{3}}{3}$

Would you like more details, or do you have any questions?

Here are 5 related questions:

1. What is the tangent function's period and how does it affect the values?
2. How do you find the reference angle for any given angle in radians?
3. What are the values of $\sin(\frac{13\pi}{6})$ and $\cos(\frac{13\pi}{6})$?
4. How do you convert radians to degrees, and what is $\frac{13\pi}{6}$ in degrees?
5. What are the properties of trigonometric functions in different quadrants?

Tip: Remember that the tangent function repeats every $\pi$ radians.