# Trigonometric functions

Trigonometric functions are functions related to an angle. There are a total of 6 trigonometric functions, 3 of which, Sine, Cosine, and Tangent, are most commonly used, while the other 3, Secant, Cosecant, and Cotangent, are used less often.

## Trigonometric function definitions

Consider an angle θ as one angle in a right triangle.

- a is the length of the side opposite the angle θ.
- b is the length of the side next to the angle θ and the right angle.
- c is the length of the side opposite the right angle. It is also the longest side.

The following are the definitions of the trigonometric functions based on the right triangle above. In mathematics, these functions are often written in their abbreviated forms.

sine: | |

cosine: | |

tangent: | |

cosecant: | |

secant: | |

cotangent: |

Example:

Given a=1, b=√3, and c=2, what are the values of sin(θ), cos(θ), and tan(θ)?

## Unit-circle definitions

Trigonometric functions can also be defined using the coordinate values of a unit circle (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 ray intersects the unit circle at point A and intersects the line x=1 at point B. The coordinate values of A and B are (x_{A}, y_{A}) and (x_{B}, y_{B}).

sin(θ) = y_{A} |

cos(θ) = x_{A} |

tan(θ) = y_{B} |

For simplicity, only the 3 major trigonometric functions are drawn and defined above. The other 3 functions can be defined on the unit circle in a similar way.

One major limitation of defining trigonometric functions using right triangles is that the angles are restricted to being between 0° and 90°. Trigonometric functions defined using the unit circle do not have such limitations, and the definitions can thus be used with any angle.

## Trigonometric function properties

The 6 trigonometric functions form many relationships. The following are some of the most important:

sin(-θ) = -sin(θ) |

cos(-θ) = cos(θ) |

tan(-θ) = -tan(θ) |

cos(90° - θ) = sin(θ) |

sin(90° - θ) = cos(θ) |

cos(90° + θ) = -sin(θ) |

sin(90° + θ) = cos(θ) |

sin(θ + 180°) = -sin(θ) |

cos(θ + 180°) = -cos(θ) |

sin(θ + n×360°) = sin(θ) |

cos(θ + n×360°) = sin(θ) |

sin^{2}(θ) + cos^{2}(θ) = 1 |

tan(θ + n×360°) = sin(θ) |

In the equations above, n is an integer.

Also, aside from converting between different trigonometric functions, one of the main uses for the equations above is converting an angle of any value into an angle value between 0° and 90° to make the numbers easier to work with.

Example:

What is the value of sin(-750°)?

sin(-750°) | = sin(2×360° - 750°) |

= sin(-30°) | |

= -sin(30°) | |

= -0.5 |

## Trigonometric function values

The values of trigonometric functions can be found through the coordinate values of the intersections on a unit circle. The trigonometric functions of many angles also have algebraic values. The following are a list of the algebraic values of some of the most commonly used angles.

The sign of trigonometric functions vary in different quadrants. The following are the signs of the 3 main trigonometric functions in each quadrant.

## Inverse trigonometric functions

The inverse trigonometric functions are the inverse functions of the trigonometric functions:

Arcsin, arccos, and arctan are the inverse functions of sine, cosine, and tangent respectively. The computation of inverse trigonometric functions results in an angle value.

Example:

Given sin(30°) = 0.5, what is arcsin(0.5)?

arcsin(0.5) = 30° |

note: this only provides the results in the first quadrant. Since arcsin is a periodic function, arcsin(0.5) is also equal to 150°, 30°+n×360°, or 150°+n×360°, where n is an integer.