Trigonometric ratios of some specific angle are defined as the ratio of the sides of a right-angled triangle with respect to any of its acute angles. Trigonometric ratios of some specific angle include 0°, 30°, 60° and 90°. In this sub-topic we will learn how to find the trigonometric ratios of 30°, 45° and 60° in detail.

The special angles with measures of 30°, 45° and 60° have a way of showing up in various applications of trigonometry. Trigonometric ratios for these special angles can easily be deduced with the help of a right-angled triangle.

Using some elementary geometry, we can find the values of sine, cosine and tangent functions of the special angles. To begin with, consider an equilateral triangle ABC having sides of length 2 units.

Recall from plane geometry that the vertex angle of an equilateral triangle all measures 60° and that the bisector

of vertex angle A is perpendicular bisector of the opposite side

Therefore,

Applying the Pythagorean theorem to right triangle ADB, we have

(AD)² + (DB)² = (AB)²
(AD)² + 1² = 2²
(AD)² + 1 = 4
(AD)² = 3
AD =√3 .