# Trigonometric Ratios

In trigonometry,** trigonometric ratios **are derived from the sides of a right-angled triangle. There are six 6 ratios such as **sine, cosine, tangent, cotangent, cosecant, and secant.** You will learn here to build a trigonometry table for these ratios for some particular angles, such as 0 °, 30 °, 45 °, 60 °, 90°. There are many trigonometry formulas and trigonometric identities, which are used to solve complex equations in geometry.

## Trigonometric Ratios Definition

It is defined as the values of all the trigonometric function based on the value of the ratio of sides in a right-angled triangle.** The ratios of sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle**. Consider a right-angled triangle, right-angled at B.

With respect to ∠C, the ratios of trigonometry are given as:

**sine:**Sine of an angle is defined as the ratio of the side opposite(perpendicular side) to that angle to the hypotenuse.**cosine:**Cosine of an angle is defined as the ratio of the side adjacent to that angle to the hypotenuse.**tangent:**Tangent of an angle is defined as the ratio of the side opposite to that angle to the side adjacent to that angle.**cosecant:**Cosecant is a multiplicative inverse of sine.**secant:**Secant is a multiplicative inverse of cosine.**cotangent:**Cotangent is the multiplicative inverse of the tangent.

The above ratios are abbreviated as sin, cos, tan, cosec, sec and tan respectively in the order they are described. So, for Δ *ABC*, the ratios are defined as:

sin ∠C = (Side opposite to ∠C)/(Hypotenuse) = AB/AC

cos ∠C = (Side adjacent to ∠C)/(Hypotenuse) = BC/AC

tan ∠C = (Side opposite to ∠C)/(Side adjacent to ∠C) = AB/AC = sin ∠C/cos ∠C

cosec ∠C= 1/sin ∠C = (Hypotenuse)/ (Side Opposite to ∠C) = AC/AB

sec∠A = 1/cos ∠C = (Hypotenuse)/ (Side Opposite to ∠C) = AC/BC

cot ∠C = 1/tan ∠C = (Side adjacent to ∠C)/(Side opposite to ∠C)= BC/AB

In right Δ ABC, if ∠*A *and ∠*C* are assumed as 30° and 60°, then there can be infinite right triangles with those specifications but all the ratios written above for ∠*C* in all of those triangles will be same. So, all the ratios for any of the acute angles (either ∠*A* or ∠*C*) will be the same for every right triangle. This means that the ratios are independent of lengths of sides of the triangle.

## Trigonometric Ratios Table

Below is the table where each ratios values are given with respect to different angles, particularly used in calculations.

Angle |
0° |
30° |
45° |
60° |
90° |

Sin∠C | 0 | 1/2 | 1/√2 | √3/2 | 1 |

Cos∠C | 1 | √3/2 | 1/√2 | 1/2 | 0 |

Tan∠C | 0 | 1/√3 | 1 | √3 | ∞ |

Cot∠C | ∞ | √3 | 1 | 1/√3 | 0 |

Sec∠C | 1 | 2/√3 | √2 | 2 | ∞ |

Cosec∠C | ∞ | 2 | √2 | 2/√3 | 1 |

### Trigonometry Applications

Trigonometry is one of the most important branches of mathematics. Some of the applications of trigonometry are:

- Measuring the heights of towers or big mountains
- Determining the distance of the shore from the sea
- Finding the distance between two celestial bodies
- Determining the power output of solar cell panels at different inclinations
- Representing different physical quantities such as mechanical waves, electromagnetic waves, etc.