### Sine definition

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

sin *α = a / c*

**Example**

a = 5″

c = 10″

sin *α = a / c* = 5 / 10 = 0.5

### Graph of sine

TBD

### Sine rules

Rule name | Rule |

Symmetry | sin(-θ) = -sin θ |

Symmetry | sin(90°- θ) = cos θ |

Pythagorean identity | sin^{2}α + cos^{2}α = 1 |

sin θ = cos θ × tan θ | |

sin θ = 1 / csc θ | |

Double angle | sin 2θ = 2 sin θ cos θ |

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

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

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

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

Law of sines | a / sin α = b / sin β = c / sin γ |

Derivative | sin' x = cos x |

Integral | ∫ sin x dx = - cos x + C |

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

### Inverse sine function

The arcsine of ** x** is defined as the inverse sine function of

**when**

*x***-1 ≤**.

*x*≤ 1When the sine of ** y** is equal to

**:**

*x*Advertisements

sin *y = x*

Then the arcsine of ** x** is equal to the inverse sine function of

**, which is equal to**

*x***:**

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

### Sine table

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

-90^{o} | -π/2 | -1 |

-60^{o} | -π/3 | -√3/2 |

-45^{o} | -π/4 | -√2/2 |

-30^{o} | -π/6 | -1/2 |

0^{o} | 0 | 0 |

30^{o} | π/6 | 1/2 |

45^{o} | π/4 | √2/2 |

60^{o} | π/3 | √3/2 |

90^{o} | π/2 | 1 |

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