### Cosine definition

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

cos *α* = *b / c*

**For example:**

*b* = 5″*c* = 10″

cos *α* = *b / c* = 5 / 10 = 0.5

### Graph of cosine

TBD

### Cosine rules

Rule name | Rule |

Symmetry | cos(-θ) = cos θ |

Symmetry | cos(90°- θ) = sin θ |

Pythagorean identity | sin2(α) + cos2(α) = 1 |

cos θ = sin θ / tan θ | |

cos θ = 1 / sec θ | |

Double angle | cos 2θ = cos2 θ - sin2 θ |

Angles sum | cos(α+β) = cos α cos β - sin α sin β |

Angles difference | cos(α-β) = cos α cos β + sin α sin β |

Sum to product | cos α + cos β = 2 cos [(α+β)/2] cos [(α-β)/2] |

Difference to product | cos α - cos β = - 2 sin [(α+β)/2] sin [(α-β)/2] |

Law of cosines | |

Derivative | cos' x = - sin x |

Integral | ∫ cos x dx = sin x + C |

Euler's formula | cos x = (e + ^{ix}e) / 2 ^{-ix} |

### Inverse cosine function

The arccosine of ** x** is defined as the inverse cosine function of

**when**

*x***-1≤**.

*x*≤1When 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:

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

**For example:**

arccos 1 = cos^{-1 }1 = 0 rad = 0°

### Cosine table

x(^{o}) | x (rad) | cos x |

180^{o} | π | -1 |

150^{o} | 5π/6 | -√3/2 |

135^{o} | 3π/4 | -√2/2 |

120^{o} | 2π/3 | -1/2 |

90^{o} | π/2 | 0 |

60^{o} | π/3 | 1/2 |

45^{o} | π/4 | √2/2 |

30^{o} | π/6 | √3/2 |

0^{o} | 0 | 1 |

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