The six **trigonometric ratios** are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec). In geometry, trigonometry is a branch of mathematics that deals with the sides and angles of a right-angled triangle. Therefore, trig ratios are evaluated with respect to sides and angles.

The trigonometry ratios for a specific angle ‘θ’ is given below:

Trigonometric Ratios | |

Sin θ | Opposite Side to θ/Hypotenuse |

Cos θ | Adjacent Side to θ/Hypotenuse |

Tan θ | Opposite Side/Adjacent Side & Sin θ/Cos θ |

Cot θ | Adjacent Side/Opposite Side & 1/tan θ |

Sec θ | Hypotenuse/Adjacent Side & 1/cos θ |

Cosec θ | Hypotenuse/Opposite Side & 1/sin θ |

Note: Opposite side is the perpendicular side and the adjacent side is the base of the right-triangle. Also, check out trigonometric functions to learn about each of these ratios or functions in detail.Trigonometric Identities

## Definition

Trigonometric Ratios are defined as the values of all the trigonometric functions 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**.

The three sides of the right triangle are:

- Hypotenuse (the longest side)
- Perpendicular (opposite side to the angle)
- Base (Adjacent side to the angle)

## How to Find Trigonometric Ratios?

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 C = 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

The trigonometric ratios for some specific angles such as 0 °, 30 °, 45 °, 60 ° and 90° are given below, which are commonly used in mathematical 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 |

From this table, we can find the value for the trigonometric ratios for these angles. Examples are:

- Sin 30° = ½
- Cos 90° = 0
- Tan 45° = 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.

It is evident from the above examples that trigonometry has its involvement in a major part of our day-to-day life and much more. In most of the applications listed above, something was being measured and that is what trigonometry is all about.

## Solved Problems

**Q.1:** If in a right-angled triangle ABC, right-angled at B, hypotenuse AC = 5cm, base BC = 3cm and perpendicular AB = 4cm and if ∠ACB = θ, then find tan θ, sin θ and cos θ.

**Sol:** Given,

In ∆ABC,

Hypotenuse, AC = 5cm

Base, BC = 3cm

Perpendicular, AB = 4cm

Then,

tan θ = Perpendicular/Base = 4/3

Sin θ = Perpendicular/Hypotenuse = AB/AC = ⅘

Cos θ = Base/Hypotenuse = BC/AC = ⅗

**Q.2:** Find the value of tan θ if sin θ = 12/5 and cos θ = ⅗.

**Sol:** Given, sin θ = 12/5 and cos θ = ⅗

As we know,

Tan θ = Sin θ/Cos θ

Tan θ = (12/5)/(⅗)

Tan θ = 12/3

Tan θ = 4

## Practice Questions

- Find the value of sin θ, if tan θ = ¾ and cos θ = ½.
- Find tan θ if sin θ = 4/3 and cos θ = 3/2
- Find sec θ, if cos θ = 9/8
- Find cosec θ, if sin θ 16/5