# Trigonometric ratios of some Specific Angles

**Trigonometry** is all about triangles or to be more precise the relationship between the angles and sides of a triangle (right-angled triangle). In this article, we will be discussing the ratio of sides of a right-angled triangle concerning its acute angle called trigonometric ratios of the angle and find the trigonometric ratios of specific angles: 0°, 30°, 45°, 60°, and 90°.

Consider the following triangle,

The side BA is opposite to the angle ∠BCA so we call BA the opposite side to ∠C and AC is the hypotenuse; the other side BC is the adjacent side to ∠C.

### Trigonometric Ratios of angle C

Sine:Sine of ∠C is the ratio of the side opposite to C (BA) to the hypotenuse (AC).

Cosine:Cosine of ∠C is the ratio of the side adjacent to C (BC) and the hypotenuse (AC).

Tangent:The tangent of ∠C is the ratio between the side opposite (BA) and adjacent to C (BC).

Cosecant:Cosecant of ∠C is the reciprocal of sin C therefore it is the ratio of the hypotenuse (AC) to the side opposite to C (BA).

Secant:Secant of ∠C is the reciprocal of cos C therefore it is the ratio of the hypotenuse (AC) to the side adjacent to C (BC).

Cotangent:Cotangent of ∠C is the reciprocal of tan C that is the ratio of the side adjacent to C (BC) to the side opposite to C (BA).

### Finding trigonometric ratios for angles 0°, 30°, 45°, 60°, 90°

Considering the length of the hypotenuse AC = a, BC = b and, BA = c.

**A. For angles 0° and 90°**

If angle A = 0°, the length of the opposite side would be zero and hypotenuse = adjacent side, and if A = 90°, the hypotenuse = opposite side. So, with the help of the above formulas for the trigonometric ratios we get –

if A = 0°

if A = 90°

Here some of the trigonometric ratios result as **not defined** as at the particular angle it is divided by 0 which is undefined.

**B. For angles 30° and 60°**

Consider an equilateral triangle ABC. Since each angle in an equilateral triangle is 60°, therefore,

∠A = ∠B = ∠C = 60°.

∆ABD is a right triangle, right-angled at D with ∠BAD = 30° and ∠ABD = 60°,

Here ∆ADB and ∆ADC are similar as they are **Corresponding parts of Congruent triangles (CPCT)**.

Now we know the values of AB, BD, and AD, So the trigonometric ratios** for angle 30°** are,

**For angle 60°**

**C. For angle 45°**

In a right-angled triangle if one angle is 45° then the other angle is also 45° thus, making it an isosceles right-angle triangle.

If the length of side BC = a then length of AB = a and length of AC(hypotenuse) is a√2 using Pythagoras Theorem, then

### All values

∠A | 0° | 30° | 45° | 60° | 90° |

sin A | 0 | 1/2 | 1/√2 | √3/2 | 1 |

cos A | 1 | √3/2 | 1/√2 | 1/2 | 0 |

tan A | 0 | 1/√3 | 1 | √3 | Not defined |

cosec A | Not defined | 2 | √2 | 2/√3 | 1 |

sec A | 1 | 2/√3 | √2 | 2 | Not defined |

cot A | Not defined | √3 | 1 | 1/√3 | 0 |