**Trigonometry Function Formulas**

**Trigonometric Function of a Right Triangle :**

sin α = a / c = opposite / hypotenuse

cos α = b / c = adjacent / hypotenuse

tan α = a / b = opposite / adjacent

cot α = b / a = adjacent / opposite

sec α = c / b Cosec α = c / a

**Basic Formula :**

sin^{2} α + cos^{2} α = 1

tan α . cot tan α = 1

tan α = sin α / cos α = 1 / cot tan α

cot tan α = cos α / sin α = 1 / tan α

1 + tan^{2} α = 1 / cos^{2} α = sec^{2} α

1 + cot tan^{2} α = 1 / sin^{2} α = cos sec^{2} α

**Trigonometr**ic Table

α | 0^{0} | 30^{0} | 45^{0} | 60^{0} | 90^{0} | 120^{0} | 180^{0} | 270^{0} | 360^{0} |

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

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

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

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

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

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

**Co-Ratios**

sin | cos | tan | cot | |

-α | -sin α | +cos α | -tan α | -cot α |

90^{0} – α | +cos α | +sin α | +cot α | +tan α |

90^{0} + α | +cos α | -sin α | -cot α | -tan α |

180^{0} – α | +sin α | -cos α | -tan α | -cot α |

180^{0} + α | -sin α | -cos α | +tan α | +cot α |

270^{0} – α | -cos α | -sin α | +cot α | +tan α |

270^{0} + α | -cos α | +sin α | -cot α | -tan α |

360^{0}k – α | -sin α | +cos α | -tan α | -cot α |

360^{0}k – α | +sin α | +cos α | +tan α | +cot α |

**Trigonometry Addition Formula:**

- sin(A + B) = sinA cosB + cosA sinB
- sin(A – B) = sinA cosB – cosA sinB

- cos(A + B) = cosA cosB – sinA sinB
- cos(A – B) = cosA cosB + sinA sinB

- tan (A + B) = tanA + tanB / 1 – tanA tanB
- tan(A – B) = tanA – tanB / 1 + tanA tanB

- cot (A+ B) = cotA cotB – 1 / cotA + cotB

**Product of Trigonometric Functions:**

- sin α cos β = 1/2 [ sin (α + β) + sin(α – β)]
- cos α cos β = 1/2 [ sin (α + β) + sin(α – β)]
- cos α cos β = 1/2 [ cos (α + β) + cos(α – β)]
- sin α sin β = 1/2 [ cos (α – β) + cos(α + β)]

- tan α tan β = tan α + tan β / cot tan α + cot tanβ = – tanα – tan β / cot tan α – cot tan β

Trigonometric Formula with t = tan(x/2)

sinx = 2t / 1 + t^{2}

cos x = 1 – t^{2} / 1 + t^{2}

tan x = 2t / 1 – t^{2}

cot x = 1 – t^{2} / 2t

**Trigonometric Relation Between Functions:**

**Angle of a Plane Triangle :**

- A, B, C are 3 angles of a triangle
- sin A + sin B + sin c = 4 cos(A / 2) cos(B/2) cos(C/2)
- cosA + cos B + cos C = 4 sin(A/2) sin(B/2) sin(C/2) + 1
- sinA + sinB – sinC = 4sin (A/2) sin (B/2) cos (C/2)

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