### Tangent definition

In a right triangle ABC the tangent of **α**, **tan(α)** is defined as the ratio betwween the side opposite to angle α and the side adjacent to the angle α:

tan *α = a / b*

**For example:**

a = 5″

b = 10″

tan *α* = *a */ *b* = 5 / 10 = 0.5

### Graph of tangent

TBD

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### Tangent rules

Rule name | Rule |

Symmetry | tan(-θ) = -tan θ |

Symmetry | tan(90°- θ) = cot θ |

tan θ = sin θ / cos θ | |

tan θ = 1 / cot θ | |

Double angle | tan 2θ = 2 tan θ / (1 - tan^{2}θ) |

Angles sum | tan(α+β) = (tan α + tan β) / (1 - tan α tan β) |

Angles difference | tan(α-β) = (tan α - tan β) / (1 + tan α tan β) |

Derivative | tan' x = 1 / cos^{2}(x) |

Integral | ∫ tan x dx = - ln |cos x| + C |

Euler's formula | tan x = (e) / i (^{ix} - e^{-ix}e) ^{ix} + e^{-ix} |

### Inverse tangent function

The arctangent of * x* is defined as the inverse tangent function of

**when**

*x**is real (*

**x****).**

*x∈ℝ*When the tangent of * y* is equal to

**:**

*x*tan *y = x*

Then the arctangent of x is equal to the inverse tangent function of x, which is equal to y:

arctan *x* = tan^{-1} *x = y*

**For example:**

arctan 1 = tan^{-1} 1 = π/4 rad = 45°

### Tangent table

x (rad) | x (°) | tan(x) |

-π/2 | -90° | -∞ |

-1.2490 | -71.565° | -3 |

-1.1071 | -63.435° | -2 |

-π/3 | -60° | -√3 |

-π/4 | -45° | -1 |

-π/6 | -30° | -1/√3 |

-0.4636 | -26.565° | -0.5 |

0 | 0° | 0 |

0.4636 | 26.565° | 0.5 |

π/6 | 30° | 1/√3 |

π/4 | 45° | 1 |

π/3 | 60° | √3 |

1.1071 | 63.435° | 2 |

1.2490 | 71.565° | 3 |

π/2 | 90° | ∞ |

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