1. **State the problem:** Evaluate $\tan\left(-\frac{\pi}{6}\right)$ without using a calculator. 2. **Recall the formula and properties:** The tangent function is odd, meaning $\tan(-x) = -\tan(x)$. Also, $\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}$. 3. **Apply the odd function property:** $$\tan\left(-\frac{\pi}{6}\right) = -\tan\left(\frac{\pi}{6}\right)$$ 4. **Substitute the known value:** $$= -\frac{1}{\sqrt{3}}$$ 5. **Simplify the expression:** Rationalize the denominator: $$-\frac{1}{\sqrt{3}} = -\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$$ 6. **Final answer:** $$\tan\left(-\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{3}$$ This quantity is defined and simplified.