Arccos(x), cos^{-1}(x), inverse cosine function.

### Arccos definition

The arccosine of x is defined as the inverse cosine function of x when -1≤x≤1.

When the cosine of y is equal to x:

cos *y = x*

Then the arccosine of x is equal to the inverse cosine function of x, which is equal to y:

arccos *x* = cos^{-1} *x = y*

Here cos^{-1} x means the inverse cosine and does not mean cosine to the power of -1.

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**For example:**

arccos 1 = cos^{-1} 1 = 0 rad = 0°

### Graph of arccos

### Arccos rules

Rule name | Rule |

Cosine of arccosine | cos( arccos x ) = x |

Arccosine of cosine | arccos( cos x ) = x + 2kπ, when k∈ℤ (k is integer) |

Arccos of negative argument | arccos(-x) = π - arccos x = 180° - arccos x |

Complementary angles | arccos x = π/2 - arcsin x = 90° - arcsin x |

Arccos sum | arccos(α) + arccos(β) = arccos( αβ - √(1-α^{2})(1-β^{2}) ) |

Arccos difference | arccos(α) - arccos(β) = arccos( αβ + √(1-α^{2})(1-β^{2}) ) |

Arccos of sin of x | arccos( sin x ) = -x - (2k+0.5)π |

Sine of arccosine | cos (arcsin x) = sin (arccos x) = √1 - x^{2} |

Tangent of arccosine | |

Derivative of arccosine | |

Indefinite integral of arccosine | ∫ arccos x dx = x arccos x - √1-x^{2} + C |

### Arccos table

x | arccos(x)(rad) | arccos(x)(°) |

-1 | π | 180° |

-√3/2 | 5π/6 | 150° |

-√2/2 | 3π/4 | 135° |

-1/2 | 2π/3 | 120° |

0 | π/2 | 90° |

1/2 | π/3 | 60° |

√2/2 | π/4 | 45° |

√3/2 | π/6 | 30° |

1 | 0 | 0° |

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