Arctan(x), tan^{-1}(x), inverse tangent function.

### Arctan definition

The arctangent of x is defined as the inverse tangent function of x when x is real (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:**

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arctan 1 = tan^{-1} 1 = π/4 rad = 45°

### Graph of arctan

*f*(*x*) = arctan(*x*)

### Arctan rules

Rule name | Rule |

Tangent of arctangent | tan (arctan x) = x |

Arctan of negative argument | arctan(-x) = - arctan x |

Arctan sum | arctan α + arctan β = arctan [(α+β) / (1-αβ)] |

Arctan difference | arctan α - arctan β = arctan [(α-β) / (1+αβ)] |

Sine of arctangent | |

Cosine of arctangent | |

Reciprocal argument | |

Arctan from arcsin | |

Derivative of arctan | |

Indefinite integral of arctan |

### Arctan table

x | arctan(x) (rad) | arctan(x) (°) |

-∞ | -π/2 | -90° |

-3 | -1.2490 | -71.565° |

-2 | -1.1071 | -63.435° |

-√3 | -π/3 | -60° |

-1 | -π/4 | -45° |

-1/√3 | -π/6 | -30° |

-0.5 | -0.4636 | -26.565° |

0 | 0 | 0° |

0.5 | 0.4636 | 26.565° |

1/√3 | π/6 | 30° |

1 | π/4 | 45° |

√3 | π/3 | 60° |

2 | 1.1071 | 63.435° |

3 | 1.2490 | 71.565° |

∞ | π/2 | 90° |

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