Trigonometry formulas are very important for solving any trigonometric questions. So we have bring up here complete list of trigonometry formulas which are used in the trigonometry questions . We have included all the formulas of trigonometry here.

## Trigonometry and Trigonometry Formulas

Trigonometry is the branch of mathematics that deals with the three side figures like triangle. A Triangle is 2 dimension figure which has 3 sides. Trigonometry helps to understand the triangles and use it to solve queries in our daily life like distance measurement, speed measurement, angle measurement, etc

### History of Trigonometry Ratios

The first use of the idea of ‘sine’ in the way we use it today was in the work Aryabhatiyam by Aryabhata, in A.D. 500. Aryabhata used the word ardha-jya for the half-chord, which was shortened to jya or jiva in due course. When the Aryabhatiyam was translated into Arabic, the word jiva was retained as it is. The word jiva was translated into sinus, which means curve, when the Arabic version was translated into Latin. Soon the word sinus, also used as sine, became common in mathematical texts throughout Europe. An English Professor of astronomy Edmund Gunter (1581–1626), first used the abbreviated

notation ‘sin’.

The origin of the terms ‘cosine’ and ‘tangent’ was much later. The cosine function arose from the need to compute the sine of the complementary angle. Aryabhatta called it **kotijya**. The name cosinus originated with Edmund Gunter. In 1674, the English Mathematician Sir Jonas Moore first used the abbreviated notation ‘cos’.

## Trigonometry Formula

So Here are all the trigonometry formulas from basics to advanced.

1. Trigonometric Ratios

2.Short Names of Trigonometric Ratio’s .

3.Trigonometry Relation with other Trigonometry ratios

4.Pythagoras Theorem

5.Trigonometry table

6.Trigonometric Ratio of Complementary Angles

7.Trigonometry Identities

8.Trigonometric Functions

9.Angles of any triangle – Laws of sine and laws of cosine

10.Inverse Trigonometric functions/formulas

### Trigonometric Ratios

For a right angled triangle we can calculate the trigonometric ratios as follows, Let us consider a right angled triangle as shown in figure . The triangles has three sides as AB, BC, CA and three angles as A,B,C . The trigonometric ratios can be found as under :-

**Note :** AB side is adjacent to angle A,**Note :** BC side is opposite to angle A, **Note : **AC side is the hypotenuse as this side is opposite to angle B (right angle 90°) .

Sine ∠A = (Side opposite to angle A ) / Hypotenuse = \(\frac{BC}{AC}\)

Cosine of ∠A = (Side adjacent to angle A)/ Hypotenuse = \(\frac{AB}{AC}\)

Tangent of ∠A =(Side Opposite to angle A )/ (Side adjacent to angle A) = \(\frac{BC}{AB}\)

cosecant of ∠A = ( Hypotenuse ) / (Side opposite to angle A) = \(\frac{AC}{BC}\)

secant of ∠A = ( Hypotenuse ) / ( Side adjacent to angle A ) = \(\frac{AC}{AB}\)

cotangent of ∠A = ( Side adjacent to angle A) / (Side opposite to angle A ) = \(\frac{AB}{BC}\)

### Short Names of Trigonometric Ratios

Here are the short names of the trigonometric ratios .

Complete name of Trigonometric Ratio | Short Name of Trigonometric Ratio | |
---|---|---|

1. | Sine θ | Sin θ |

2. | Cosine θ | Cos θ |

3. | Tangent θ | Tan θ |

4. | Cosecant θ | Cosec θ |

5. | Secant θ | Sec θ |

6. | Cotangent θ | Cot θ |

Trigonometric ratios in the form of side of a triangle . Let the angle is θ (theta), and P = Perpendicular of the triangle with respect to angle θ and B= Base of the triangle with respect to angle θ.

Sin θ = \(\frac{P}{H}\)

Cos θ = \(\frac{B}{H}\)

Tan θ = \(\frac{P}{B}\)

Cosec θ = \(\frac{H}{P}\)

Sec θ = \(\frac{H}{B}\)

Cot θ = \(\frac{B}{P}\)

### Pythagoras Theorem

Pythagoras theorem: For a right angle triangle as given below, the square of the length of the hypotenuse is always equal to the sum of the squares of other two side.

(CA)^{2} = (AB)^{2} + (BC)^{2}

H^{2} = P^{2} + B^{2}

## Trigonometry Table

Trigonometry table is used to get the values of the trigonometric ratio’s at a particular angle which are defined in the table. using the Trigonometry Table , we can easily calculate the the values of trigonometric ratios at angles – 0°, 30° , 45° , 60°, 90° , 180°, 270°, 360°

**Trigonometry Table 1** : Angle are from 0° to 90° [ Angles are in degree ] – first three trigonometric ratio

∠A | 0° | 30° | 45° | 60° | 90° |
---|---|---|---|---|---|

Sin ∠A | 0 | \( \frac{1}{2} \) | \( \frac{1}{√2} \) | \( \frac{√3}{2} \) | 1 |

Cos ∠A | 1 | \( \frac{√3}{2} \) | \( \frac{1}{√2} \) | \( \frac{1}{2} \) | 0 |

tan ∠A | 0 | \( \frac{1}{√3} \) | 1 | √3 | Not Defined |

**Trigonometry Table 2** : Angle are from 0° to 360° [ Angles are in degree ] – All trigonometric ratios

∠θ | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
---|---|---|---|---|---|---|---|---|

Sin ∠θ | 0 | \( \frac{1}{2} \) | \( \frac{1}{√2} \) | \( \frac{√3}{2} \) | 1 | 1 | -1 | 0 |

Cos ∠θ | 1 | \( \frac{√3}{2} \) | \( \frac{1}{√2} \) | \( \frac{1}{2} \) | 0 | -1 | 0 | -1 |

tan ∠θ | 0 | \( \frac{1}{√3} \) | 1 | √3 | ∞ | 0 | -∞ | 0 |

Cosec θ | ∞ | 2 | √2 | \( \frac{2}{√3} \) | 1 | 1 | -1 | ∞ |

Sec ∠θ | 1 | \( \frac{2}{√3} \) | √2 | 2 | ∞ | -1 | 0 | -1 |

Cot ∠θ | ∞ | √3 | 1 | \( \frac{1}{√3} \) | 0 | ∞ | 0 | ∞ |

**Trigonometry Table **3 : Angle are from 0 to 2π [ Angles are in radian ]

∠θ | 0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
---|---|---|---|---|---|---|---|---|

Sin ∠θ | 0 | \( \frac{1}{2} \) | \( \frac{1}{√2} \) | \( \frac{√3}{2} \) | 1 | 1 | -1 | 0 |

Cos ∠θ | 1 | \( \frac{√3}{2} \) | \( \frac{1}{√2} \) | \( \frac{1}{2} \) | 0 | -1 | 0 | -1 |

tan ∠θ | 0 | \( \frac{1}{√3} \) | 1 | √3 | ∞ | 0 | -∞ | 0 |

Cosec θ | ∞ | 2 | √2 | \( \frac{2}{√3} \) | 1 | 1 | -1 | ∞ |

Sec ∠θ | 1 | \( \frac{2}{√3} \) | √2 | 2 | ∞ | -1 | 0 | -1 |

Cot ∠θ | ∞ | √3 | 1 | \( \frac{1}{√3} \) | 0 | ∞ | 0 | ∞ |

### Trigonometric Ratio relation with each other

All the six trigonometric ratio’s are related to each other by reciprocal as follows :

Sin θ = \(\frac{1}{Cosec \ θ }\)

Cos θ = \(\frac{1}{Sec\ θ }\)

Tan θ = \(\frac{1}{Cot\ θ}\)

Cosec θ = \(\frac{1}{Sin \ θ}\)

Sec θ = \(\frac{1}{Cos \ θ }\)

Cot θ = \(\frac{1}{tan\ θ }\)

Tan θ = \(\frac{Sin\ θ }{Cos \ θ }\)

Cot θ = \(\frac{Cos\ θ}{Sin\ θ}\)

### Trigonometric Ratio of Complementary/Supplementary Angles

Here are all the trigonometry formulas of the complementary and supplementary angles.**A. Trigonometry formulas of Complementary angles**

1. Cos ( \(\frac{π}{2}\) – x ) = Sin x

2. Cos (\(\frac{π}{2}\) + x ) = – Sin x

3. Sin (\(\frac{π}{2}\) + x ) = Cos x

4. Sin (\(\frac{π}{2}\) – x ) = Cos x

**B. Trigonometry formulas of Supplementary angles**

1. Sin (π – x) = Sin x

2. Sin (π + x )= – Sin x

3. Cos (π+ x ) = – cos x

4. Cos (π – x ) = -cos x

### Trigonometric identities

Trigonometric identities are the relations between all the trigonometric ratio of one or more than one angle. There are a total of three trigonometric identities All the trigonometric identities are provided below :-

a) Sin^{2}A +Cos^{2}A = 1

b) 1 + tan^{2}A = Sec^{2}A – for 0 ≤ A < 90

c) 1+ cot^{2}A = Cosec^{2}A – for 0 < A ≤ 90

### Trigonometry Formulas of Angle in the form of degree and radians

Some time we use the angles in the form of degree, but some questions are solved by the radian angle. So here are all the trigonometry formula for angles in the form of degree and radian.

**Sexagesimal System (degree Measure)**

1 right angle = 90 degree (written as 90°)

1 degree (1°) = 60 minutes ( written as 60′)

1 minute (1′) = 60 seconds (written as 60″)

**Centesimal System **

1 right angle = 100 grades ( written as 100^{g})

1 grade (1^{g}) = 100 minutes (written as 100′)

1 minute (1′) = 100 minutes (written as 100″ )

**Circular System **

1 radian = \(\frac{2}{π}\) rt ∠s

**Relation Between Degree and Radian **

1 radian = \(\frac{180°}{π}\) ⇒ π radian = 180°

1 degree = \(\frac{π}{180°}\) ⇒ 180° = π radian

**Arc angle Relation**

θ = \(\frac{l}{r}\)

### Trigonometric functions

Here are the trigonometry formulas related to sum and product of two angles of the trigonometric ratios

Sin ( x + y ) = Sin x cos y + cos x cos y

Sin ( x – y ) = Sin x cos y – cos x cos y

Sin ( x + x ) = Sin ( 2x ) = 2 sin x cos x

Cos ( x + y ) = Cos x cos y – sin x sin y

Cos ( x – y ) = Cos x cos y + sin x sin y

Cos ( x + x ) = Cos (2x) = Cos^{2}x -Sin^{2}x = 2Cos^{2}x – 1 = 1- 2Sin^{2}x

Tan ( A+B) = \( \frac{ \ ( \ tan \ A \ + \ tan \ B \ ) }{ \ ( \ 1 \ – \ tan \ A \ tan \ B \ ) \ } \)

Tan (A-B) = \(\frac{\tan{A}\ -\ \tan{B}}{1+\tan{A}\tan{B}} \)

Tan ( \( \frac{π}{4} \) + x ) = \( \frac{( \ 1 \ + \ tan \ x \ ) \ }{ \ ( \ 1 \ – \ tan \ x \ )} \)

Tan ( \( \frac{π}{4} \) – x ) = \( \frac{ \ ( \ 1 \ – \ tan \ x \ ) \ }{ \ ( \ 1 \ + \ tan \ x \ ) \ } \)

Tan ( 2 A ) = \( \frac{ \ 2 \ tan \ A \ }{ \ ( \ 1 \ – \ tan^2 \ A \ ) \ } \)

Cot ( A + B ) = \( \ \frac{ \ ( \ Cot \ A \ Cot \ B \ – \ 1 \ ) \ }{ \ ( \ Cot \ A \ + \ Cot \ B \ ) \ } \)

Cot ( A – B ) = \( \ \frac{ \ (\ Cot \ A \ Cot \ B \ + \ 1 \ )\ }{ \ ( \ Cot \ B \ – \ Cot \ A \ ) \ } \)

Sin 3A = 3 Sin A – 4 sin^{3} A

Cos 3A = 4 Cos^{3} A – 3 Cos A

Tan 3A = \( \ \frac{ \ ( \ 3 \ tan \ A \ – \ tan^3 \ A \ ) \ }{ \ ( \ 1 \ – \ 3 \ tan^2 \ A \ ) \ } \)

Sin (-x) = – sin x

Cos (-x) = Cos x

Tan (-x) = – tan x

### Trigonometry formula of products of trigonometric ratios

The trigonometry formula which comprises of product of two ratio are also useful in solving the trigonometry questions . Here are the trigonometry formulas for product of trigonometry ratios.

- 2 cos x cos y = cos(x – y ) + cos ( x + y )
- 2 sin x sin y = cos ( x – y ) – cos ( x+ y )
- 2 sin x sin y = sin (x + y ) + sin ( x – y )
- 2 cos x sin y = sin (y + x ) + sin ( y – x ) = sin (x + y ) – sin ( x – y )
- e
^{it}= cos t + i sin t