# Sine Calculator

## Sine calculation

## Inverse sine calculator

### The Sine function ( sin(x) )

The sine function, denoted as **sin(x)**, is a fundamental trigonometric function that describes the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. It is a periodic function and is very important in the fields of science, engineering, signal processing, and many areas of mathematics.

The value of sin(x) oscillates between -1 and 1, and its graph forms a wave-like pattern. The sine function is defined for all real numbers, and its most important properties include:

**Periodicity**: sin(x) is a periodic function with a period of 2π radians (or 360 degrees), which means sin(x) = sin(x + 2kπ) for any integer k.**Symmetry**: It is an odd function, which implies that sin(-x) = -sin(x).**Range**: The output of the sine function is in the interval (-1, 1).

In a unit circle representation, where a circle has a radius of 1 and is centered at the origin of a Cartesian coordinate system, sin(x) corresponds to the y-coordinate of a point on the circle's circumference, where x is the angle (in radians) formed by a line connecting the point and the origin, and the positive x-axis.

#### What is Degrees (deg °) and Radians (rad) ?

In the context of trigonometric functions and other mathematical applications, "Degrees" and "Radians" refer to two different units for measuring angles:

- "Degrees" are a measure of angle using the familiar system where a complete circle is divided into 360 degrees.
- "Radians" are a measure of angle used in mathematics, especially in trigonometry and calculus, where a complete circle corresponds to 2π radians. One radian is defined as the angle created by an arc that is equal in length to the radius of the circle.

To convert between degrees and radians, the following two formulas can be used:

- From degrees to radians:

radians = degrees ×π180 - From radians to degrees:

degrees = radians ×180π

#### Table of common sine values

Angle (°) | Angle (Radians) | sin(angle) | sin(angle) |
---|---|---|---|

0° | 0 | 0 | 0.00 |

30° | π/6 | 1/2 | 0.50 |

45° | π/4 | √2/2 | 0.7071 |

60° | π/3 | √3/2 | 0.8660 |

90° | π/2 | 1 | 1.00 |

120° | 2π/3 | √3/2 | 0.8660 |

135° | 3π/4 | √2/2 | 0.7071 |

150° | 5π/6 | 1/2 | 0.50 |

180° | π | 0 | 0.00 |

210° | 7π/6 | -1/2 | -0.50 |

225° | 5π/4 | -√2/2 | -0.7071 |

240° | 4π/3 | -√3/2 | -0.8660 |

270° | 3π/2 | -1 | -1.00 |

300° | 5π/3 | -√3/2 | -0.8660 |

315° | 7π/4 | -√2/2 | -0.7071 |

330° | 11π/6 | -1/2 | -0.50 |

360° | 2π | 0 | 0.00 |

See also