arcsin(*x*), sin^{-1}(*x*), inverse sine function.

### Arcsin definition

The arcsine of x is defined as the inverse sine function of *x* when -1≤x≤1.

When the sine of y is equal to *x*:

sin *y = x*

Then the arcsine of x is equal to the inverse sine function of x, which is equal to y:

arcsin *x* = sin^{-1} *x = y*

**For example:**

arcsin 1 = sin^{-1} 1 = π/2 rad = 90°

### Graph of arcsin, *f*(*x*) = asin(*x*)

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### Arcsin rules

Rule name | Rule |

Sine of arcsine | sin(arcsin x) = x |

Arcsine of sine | arcsin(sin x) = x+2kπ, when k∈ℤ (k is integer) |

Arcsin of negative argument | arcsin(-x) = - arcsin x |

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

Arcsin sum | arcsin α + arcsin(β) = arcsin(α√(1-β^{2}) + β√(1-α^{2})) |

Arcsin difference | arcsin α - arcsin(β) = arcsin(α√(1-β^{2}) - β√(1-α^{2})) |

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

Tangent of arcsine | |

Derivative of arcsine | |

Indefinite integral of arcsine |

### Arcsin table

x | arcsin(x) | arcsin(x) (°) |

-1 | -π/2 | -90° |

-√3/2 | -π/3 | -60° |

-√2/2 | -π/4 | -45° |

-1/2 | -π/6 | -30° |

0 | 0 | 0° |

1/2 | π/6 | 30° |

√2/2 | π/4 | 45° |

√3/2 | π/3 | 60° |

1 | π/2 | 90° |

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