What is the exact value of tan 30°?

The exact value of tan 30° is \(\frac{1}{\sqrt{3}}\) or \(\frac{\sqrt{3}}{3}\).

To understand why this is the case, let's delve into some trigonometry basics. The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. For 30°, we can use a special right-angled triangle known as the 30-60-90 triangle.

In a 30-60-90 triangle, the sides are in a specific ratio: the side opposite the 30° angle is half the hypotenuse, and the side opposite the 60° angle is \(\sqrt{3}\) times the shorter side. If we consider the hypotenuse to be 2 units, the side opposite the 30° angle (the shorter side) will be 1 unit, and the side opposite the 60° angle (the longer side) will be \(\sqrt{3}\) units.

Using the definition of tangent:
\[ \tan 30° = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{\sqrt{3}} \]

To rationalise the denominator, we multiply the numerator and the denominator by \(\sqrt{3}\):
\[ \tan 30° = \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \]

So, \(\tan 30°\) can be expressed as \(\frac{1}{\sqrt{3}}\) or \(\frac{\sqrt{3}}{3}\), both of which are exact values. This understanding is crucial for solving various trigonometric problems and is a fundamental part of GCSE Maths.