## Trigonometry Formulas:

### Trigonometric Ratios Formulas:

sin θ = Perpendicular/Hypotenuse = MP/OP = y/r |

cos θ = Adjacent Side or Base/Hypotenuse = OM/OP = x/r |

tan θ = Perpendicular/Adjacent Side or Base = MP/OM = y/x |

cosec θ = Hypotenuse/Perpendicular = OP/MP = r/y |

sec θ = Hypotenuse/Adjacent Side or Base = OP/OM = r/x |

cot θ = Adjacent Side or Base/Perpendicular = OM/MP =x/y |

### Relation Between Trigonometric Ratios:

sin θ = 1/cosec θ |

cos θ = 1/sec θ |

tan θ = sin θ/cos θ |

cosec θ = 1/sin θ |

sec θ = 1/cos θ |

cot θ = cos θ/sin θ |

### Trigonometric Identities:

sin^{2} θ + cos^{2} θ = 1 |

1 + tan^{2} θ = sec^{2} θ |

1 + cot^{2} θ = cosec^{2} θ |

### Trigonometry Table:

Radian Measure = (π/180) x Degree Measure |

Degree Measure = (180/π) x Radian Measure |

Angles (In Radians) | 0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |

Angles (In Degrees) | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |

sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | 0 | -1 | 0 |

cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | -1 | 0 | 1 |

tan | 0 | 1/√3 | 1 | √3 | ∞ | 0 | ∞ | 0 |

cot | ∞ | √3 | 1 | 1/√3 | 0 | ∞ | 0 | ∞ |

sec | 1 | 2/√3 | √2 | 2 | ∞ | -1 | ∞ | 1 |

cosec | ∞ | 2 | √2 | 2/√3 | 1 | ∞ | -1 | ∞ |

### Trigonometric Functions:

I | II | III | IV | |

sin θ | +ve | +ve | -ve | -ve |

cos θ | +ve | -ve | -ve | +ve |

tan θ | +ve | -ve | +ve | -ve |

cosec θ | +ve | +ve | -ve | -ve |

sec θ | +ve | -ve | -ve | +ve |

cot θ | +ve | -ve | +ve | -ve |

sin (π/2 – θ) = cos θ | sin (π/2 + θ) = cos θ |

cos (π/2 – θ) = sin θ | cos (π/2 + θ) = – sin θ |

tan (π/2 – θ) = cot θ | tan (π/2 + θ) = – cot θ |

cot (π/2 – θ) = tan θ | cot (π/2 + θ) = – tan θ |

sec (π/2 – θ) = cosec θ | sec (π/2 + θ) = – cosec θ |

cosec (π/2 – θ) = sec θ | cosec (π/2 + θ) = sec θ |

sin (π – θ) = sin θ | sin (π + θ) = – sin θ |

cos (π – θ) = – cos θ | cos (π + θ) = – cos θ |

tan (π – θ) = – tan θ | tan (π + θ) = tan θ |

cot (π – θ) = – cot θ | cot (π + θ) = cot θ |

sec (π – θ) = – sec θ | sec (π + θ) = – sec θ |

cosec (π – θ) = cosec θ | cosec (π + θ) = – cosec θ |

sin (-x) = – sin x | cos (-x) = cos x |

### Addition and Subtraction Formulas in Trigonometry:

sin (A + B) = sin A cos B + cos A sin B |

sin (A – B) = sin A cos B – cos A sin B |

cos (A + B) = cos A cos B – sin A sin B |

cos (A – B) = cos A cos B + sin A sin B |

tan (A + B) = (tan A + tan B)/(1 – tan A tan B) |

tan (A – B) = (tan A – tan B)/(1 + tan A tan B) |

cot (A + B) = (cot A cot B – 1)/(cot B + cot A) |

cot (A – B) = (cot A cot B + 1)/(cot B – cot A) |

sin (A + B). sin (A- B) = sin^{2} A – sin^{2} B |

cos (A + B). cos (A- B) = cos^{2} A – sin^{2} B = cos^{2} B – sin^{2} A |

sin (A + B + C) = cos A cos B cos C (tan A + tan B + tan C – tan A tan B tan C) |

cos (A + B + C) = cos A cos B cos C (1- tan A tan B – tan B tan C – tan C tan A) |

tan (A + B + C) = (tan A + tan B + tan C – tan A tan B tan C)/(1- tan A tan B – tan B tan C – tan C tan A) |

### Sum and Difference of Sines or Cosines:

2 sin A cos B = sin (A + B) + sin (A – B) |

2 cos A sin B = sin (A + B) – sin (A – B) |

2 cos A cos B = cos (A + B) + cos (A – B) |

2 sin A sin B = cos (A- B) – cos (A + B) |

sin C + sin D = 2 sin (C + D)/2 . cos (C – D)/2 |

sin C – sin D = 2 cos (C + D)/2 . sin (C – D)/2 |

cos C + cos D = 2 cos (C + D)/2 . cos (C – D)/2 |

cos C – cos D = 2 sin (C + D)/2 . sin (D – C)/2 |

### Trigonometrical Ratios of Multiple Angles:

sin 2A = 2 sin A cos A = 2 tan A/(1 + tan^{2} A) |

cos 2A = cos^{2} A – sin^{2} A = 2 cos^{2} A – 1 = 1 – 2 sin^{2} A = (1 – tan^{2} A)/(1 + tan^{2} A) |

tan 2A = 2 tan A/(1 – tan^{2} A) |

cot 2A = (cot^{2} A – 1)/2cot A |

sin 3A = 3 sin A – 4 sin^{3} A |

cos 3A = 4 cos^{3} A – 3 cos A |

tan 3A = (3 tan A – tan^{3} A)/(1 – 3 tan^{2} A) |

### Sine Formula:

In any triangle ABC, a/sin A = b/sin B = c/sin C |

### Cosine Formula:

cos A = (b^{2} + c^{2} – a^{2})/2bc |

cos B = (c^{2} + a^{2} – b^{2})/2ca |

cos C = (a^{2} + b^{2} – c^{2})/2ab |

### Projection Formula:

a = b cos C + c cos B |

b = c cos A + a cos C |

c = a cos B + b cos A |