Solution: By definition, $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$. Substituting the given values, $\tan 30^\circ = \frac{1}{\sqrt{3}}$. Rationalizing the denominator, $\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$. Thus, $\boxed{\dfrac{\sqrt{3}}{3}}$.

["Understanding the Exact Value of $\ an 30^\circ$: Rationalized Form Explained", "Trigonometric functions are fundamental in mathematics, physics, and engineering, serving as the backbone for analyzing angles in triangles, waves, and rotations. One key concept is the tangent function, defined as the ratio of the opposite side to the adjacent side in a right triangle:\n[\n\ an \ heta = \frac{\ ext{opposite}}{\ ext{adjacent}}\n]\nFor a $30^\circ$ angle, this ratio is precisely $\ an 30^\circ = \frac{1}{\sqrt{3}}$. But why is it important to rationalize the denominator, and what does $\boxed{\dfrac{\sqrt{3}}{3}}$ mean practically?", "Why Rationalize the Denominator?\nIn mathematics, rationalizing the denominator means transforming an expression so that no radicals appear in the denominator. This process enhances clarity, standardization, and simplifies further calculations. For $\frac{1}{\sqrt{3}}$, rationalizing involves multiplying both numerator and denominator by $\sqrt{3}$, turning the expression into a cleaner form.", "Starting from:\n[\n\ an 30^\circ = \frac{1}{\sqrt{3}}\n]\nMultiply numerator and denominator by $\sqrt{3}$:\n[\n\frac{1 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{3}\n]\nThus,\n[\n\boxed{\dfrac{\sqrt{3}}{3}}\n]\nThis rationalized form is preferred in both academic and professional settings because it adheres to conventional mathematical presentation.", "The Broader Significance of $\ an 30^\circ = \frac{\sqrt{3}}{3}$\nThe value $\frac{\sqrt{3}}{3}$ appears frequently in trigonometric tables, calculus, geometry proofs, and physics formulas—particularly in problems involving equilibrium, vectors, and wave behavior. Having it in rationalized form ensures smoother arithmetic operations when combining expressions, solving equations, or calculating limits.", "In summary, knowing $\ an 30^\circ$ as $\frac{\sqrt{3}}{3}$ goes beyond simple substitution—it reflects mastery of clean, precise mathematical communication. Whether studying triangles or advanced science, clarity in representation strengthens understanding and reduces error. Embrace rationalized trigonometric values to elevate your problem-solving accuracy."]