1. **Restating the problem:** Find the exact value of $\tan\left(\frac{\pi}{3}\right)$.\n\n2. **Formula used:** $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. This means tangent of an angle is the ratio of sine to cosine of that angle.\n\n3. **Step-by-step explanation:**\n- The angle $\frac{\pi}{3}$ radians equals 60 degrees.\n- From the unit circle, $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$ and $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$.\n- Substitute these values into the tangent formula: $$\tan\left(\frac{\pi}{3}\right) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}.$$\n- Dividing fractions means multiplying by the reciprocal, so: $$\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{2} \times \frac{2}{1} = \sqrt{3}.$$\n\n4. **Summary:** The tangent of $\frac{\pi}{3}$ is $\sqrt{3}$ because it is the ratio of sine to cosine at that angle, and the division simplifies nicely.