What is the exact value of tan 60°?

The exact value of tan 60° is √3.

To understand why tan 60° equals √3, 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 60°, we can use an equilateral triangle, where all angles are 60° and all sides are equal.

If we split this equilateral triangle in half, we create two 30-60-90 right-angled triangles. In such a triangle, the sides have a specific ratio: the side opposite the 30° angle is half the hypotenuse, the side opposite the 60° angle is √3 times the shorter side, and the hypotenuse is twice the shorter side.

Let's assume the hypotenuse is 2 units. Therefore, the side opposite the 30° angle (shorter side) is 1 unit, and the side opposite the 60° angle (longer side) is √3 units. Using the definition of tangent:

\[ \tan(60°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{3}}{1} = \sqrt{3} \]

Thus, the exact value of tan 60° is √3. This value is crucial in trigonometry and often appears in various mathematical problems and applications. Understanding these fundamental ratios helps in solving more complex trigonometric equations and in analysing geometric shapes.