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

## 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.** Â 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 âˆ 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 Î” *ABC*, the ratios are defined as:

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

cos CÂ = (Side adjacent toÂ âˆ C)/(Hypotenuse)Â = BC/AC

tan CÂ = (Side opposite to âˆ C)/(Side adjacent to âˆ C)Â = AB/ACÂ = sin âˆ C/cos âˆ C

cosec C= 1/sin C = (Hypotenuse)/ (Side Opposite toÂ âˆ C)Â =Â AC/AB

sec CÂ = 1/cos C = (Hypotenuse)/ (Side Opposite toÂ âˆ C)Â = AC/BC

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

In rightÂ Î” *ABC,*Â if âˆ *AÂ *and âˆ *C*Â are assumed as 30Â°Â and 60Â°, then there can be infinite right triangles with those specifications but all the ratios written above for âˆ *C*Â in all of those triangles will be same. So, all the ratios for any of the acute angles (either âˆ *A*Â or âˆ *C*) will be the same for every right triangle. This means that the ratios are independent of lengths of sides of the triangle. Also, check out trigonometric functions to learn about each of these ratios or functions in detail.

## Trigonometric Ratios Table

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

Angle |
0Â°Â Â Â Â Â |
30Â°Â Â Â |
45Â°Â Â Â |
60Â°Â Â |
90Â°Â Â |

Sin C | 0 | 1/2 | 1/âˆš2 | âˆš3/2 | 1 |

Cos C | 1 | âˆš3/2 | 1/âˆš2 | 1/2 | 0 |

Tan C | 0 | 1/âˆš3 | 1 | âˆš3 | âˆž |

Cot C | âˆž | âˆš3 | 1 | 1/âˆš3 | 0 |

Sec C | 1 | 2/âˆš3 | âˆš2 | 2 | âˆž |

Cosec C | âˆž | 2 | âˆš2 | 2/âˆš3 | 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.

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Tnks for this! Very informative!