What is the tangent of a 60-degree angle?

The tangent of a 60-degree angle is √3.

In trigonometry, the tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side. For a 60-degree angle, this ratio is √3. This value is derived from the properties of an equilateral triangle, where all angles are 60 degrees, and when split into two right-angled triangles, the sides follow a specific ratio.

To understand this better, consider an equilateral triangle with each side of length 2 units. When you draw a height from one vertex to the midpoint of the opposite side, you create two right-angled triangles. Each of these right-angled triangles will have angles of 30 degrees, 60 degrees, and 90 degrees. The side opposite the 60-degree angle (the height) will be √3 units, and the side adjacent to the 60-degree angle (half the base) will be 1 unit. Therefore, the tangent of 60 degrees, which is the ratio of the opposite side to the adjacent side, is √3.

This value is important in various mathematical applications, including solving problems involving right-angled triangles and in trigonometric functions. Remembering that tan(60°) = √3 can help you quickly solve many trigonometric problems without needing to use a calculator.