# Sine

In mathematics, the sine function is one of the three main trigonometric functions. The sine of an acute angle can be defined with a right triangle: the sine value of an angle is the ratio of the length of the side that is opposite that angle (a in the diagram) to the length of the longest side of the triangle (the hypotenuse, c in the diagram). The sine function is often abbreviated as "sin."

## Unit-circle definition

The sine function can also be defined using a unit circle, which is a circle with radius one.

In a plane with a unit circle centered at the origin of a coordinate system, a ray from the origin forms an angle θ with respect to the x-axis. The y coordinate of the point at which the ray intersects the unit circle is the sine value of the angle. This definition works for any angle, not just the acute angles of right triangles.

Example:

Given a=3 and c=5, what is the value of sin(θ)?

Example:

A wheel chair ramp has an angle of 30° and a rise of 3 feet, what is the length of the ramp?

## Sine function properties

The sine function has various properties that allow the simplification of larger angles, or even the conversion between the sine and cosine function. The following are some of the common properties and conversions of the sine function. One of the main reasons for converting trigonometric functions is to convert an angle of any value into an angle value between 0° and 90° to make the numbers easier to work with.

sin(-θ) = -sin(θ) |

sin(180° + θ) = -sin(θ) |

sin(180° - θ) = sin(θ) |

sin(θ + n × 360°) = sin(θ) |

cos(90° - θ) = sin(θ) |

sin(90° - θ) = cos(θ) |

cos(90° + θ) = -sin(θ) |

sin(90° + θ) = cos(θ) |

Example:

Given sin(30°) = 0.5, what is the value of sin(150°)?

sin(150°) | = sin(180° - 150°) |

= sin(30°) = 0.5 |

Example:

Given sin(30°) = 0.5, what is the value of sin(570°)?

sin(570°) | = sin(570° - 2 x 360°) |

= sin(-150°) | |

= -sin(- 150° + 180°) | |

= -sin(30°) = -0.5 |

## Values of the sine function

Based on its definition, the value of the sine function is always between -1 and 1. The sine value of an angle between 0° and 180° is positive, and the value is negative if the angle is between 180° and 360°. In other words, the sine value is positive for an angle in the first and second quadrants of a Cartesian coordinate system. The value is negative for the third and fourth quadrants.

There are many methods that can be used to determine sine values such as referencing a table of sines, using a calculator, and approximating using the Taylor Series of sine. The following is a list of some of the important sine values.

Angle in degrees | Angle in radians | Sine value | Sine value in decimals |
---|---|---|---|

0° | 0 | 0 | 0 |

15° | 0.259 | ||

30° | 0.5 | ||

45° | 0.707 | ||

60° | 0.866 | ||

75° | 0.966 | ||

90° | 1 | 1 | |

180° | π | 0 | 0 |

270° | -1 | -1 | |

360° | 2π | 0 | 0 |

### Sine calculator:

The following is a calculator to find out either the sine value of an angle or the angle from the sine value.

### Sine wave

As the angle changes, the sine value oscillates between -1 and 1. When the sine value is plotted against its respective angle, it will form a wave shape like below, called a sine wave.

## Law of sines

The law of sines, shown below, is an equation revealing the relationship between the lengths of the sides and the sines of the angles of any triangle:

In this equation, a, b, and c are the length of the 3 sides of the triangle and A, B, and C are the 3 angles opposite each respective side.

Example:

Given side a=5, angle A=45° and angle B=75°, what are side b, side c, and angle C?

C = 180° - A - B = 180° - 45° - 75° = 60° |

## Inverse sine function

The sine function is used to determine a ratio from an angle, while the inverse sine function, denoted arcsin, is used to determine an angle from a ratio, as defined below:

Arcsin is often written as sin^{-1} in mathematics.

Example:

What is arcsin(0.5) in a right triangle?

arcsin(0.5) = 30°

Example:

Given side a=8, side c = 5, and angle C=30°, what is angle A?

C = arcsin(0.8) = 53.1° or 126.9° |

See also cosine, tangent, unit circle, trigonometric functions, trigonometry.