# Tangent Calculator

## Tangent calculation

## Inverse tangent calculator

### The Tangent function (tan(x))

The tangent function, often denoted as tan(x), is one of the six fundamental trigonometric functions. It relates the angles of a right triangle to the ratios of two of its sides. Specifically, for a right-angled triangle, the tangent of one of the non-right angles (let's call it θ) is the ratio of the length of the opposite side to the length of the adjacent side.

Here are some key points about the tangent function:

**Periodicity**: tan(x) is periodic with a period of π radians (or 180 degrees), which means tan(x) = tan(x + kπ) for any integer k.**Asymptotes:**Since cos(x) appears in the denominator of the tan(x) function, whenever cos(x) = 0, tan(x) will have vertical asymptotes. This occurs at x = (2n+1)π/2 for any integer n. At these points, the tangent function is undefined and its graph will have vertical lines (asymptotes) that the function approaches but never crosses or reaches.**Symmetry:**The function tan(x) is an odd function, which means it has rotational symmetry about the origin. In other words, tan(-x) = -tan(x).**Range:**The range of tan(x) is all real numbers, from negative infinity to positive infinity.**Graph:**The graph of tan(x) shows a repeating pattern of curves that extend vertically to infinity at the locations of the vertical asymptotes and pass through the origin.

The tangent function is widely used in various fields, including physics, engineering, and mathematics, particularly in situations involving periodic phenomena, wave motion, and oscillations. It is also a fundamental aspect of trigonometry and calculus, especially in the study of derivatives and integrals of trigonometric functions.

#### 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 tangent values

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

0° | 0 | 0 | 0.000 |

30° | π/6 | 1/√3 or √3/3 | 0.577 |

45° | π/4 | 1 | 1.000 |

60° | π/3 | √3 | 1.732 |

90° | π/2 | Undefined | - |

120° | 2π/3 | -√3 | -1.732 |

135° | 3π/4 | -1 | -1.000 |

150° | 5π/6 | -1/√3 or -√3/3 | -0.577 |

180° | π | 0 | 0.000 |

210° | 7π/6 | 1/√3 or √3/3 | 0.577 |

225° | 5π/4 | 1 | 1.000 |

240° | 4π/3 | √3 | 1.732 |

270° | 3π/2 | Undefined | - |

300° | 5π/3 | -√3 | -1.732 |

315° | 7π/4 | -1 | -1.000 |

330° | 11π/6 | -1/√3 or -√3/3 | -0.577 |

360° | 2π | 0 | 0.000 |

Note that the tangent of 90° and 270° is undefined, which is why the decimal representation is marked with a dash (-). The actual values of the tangent function can get very large in magnitude near these undefined points, going towards positive or negative infinity, depending on the direction of approach.

See also