The exact value of $\tan 30^\circ$ is given by either $\frac{1}{\sqrt{3}}$ or $\frac{\sqrt{3}}{3}$.

To grasp why this is true, we need to explore some fundamental concepts of trigonometry. The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to it. For an angle of $30^\circ$, we can utilize a special type of right triangle known as a 30-60-90 triangle.

In a 30-60-90 triangle, the lengths of the sides adhere to a specific ratio: the side opposite the $30^\circ$ angle is half the length of the hypotenuse, while the side opposite the $60^\circ$ angle is $\sqrt{3}$ times the length of the shorter side. If we set the hypotenuse to be $2$ units, the side opposite the $30^\circ$ angle (the shorter side) will measure $1$ unit, and the side opposite the $60^\circ$ angle (the longer side) will measure $\sqrt{3}$ units.

Using the definition of tangent, we can express $\tan 30^\circ$ as follows:

To rationalize the denominator, we multiply both the numerator and the denominator by $\sqrt{3}$:

Thus, $\tan 30^\circ$ can be represented as either $\frac{1}{\sqrt{3}}$ or $\frac{\sqrt{3}}{3}$, with both forms being exact values. This understanding is essential for tackling a variety of trigonometric problems and is a fundamental component of GCSE Mathematics.