# Arctan equals inverse tan

## Arctangent

In this article, we'll do the**Arctangent**and deal with it in detail. Among other things, we explain the**properties** of the arctangent and name its most important**Function values**. We'll also show you how to do this too**calculate approximately** and the function in a **Develop series** can. Finally we'll show you how**Antiderivative and derivative** of the arctangent.

In order to be able to convey the topic to you even more clearly, we also have an extra one **Video ** created for you.

### Arc tangent simply explained

If you are supposed to calculate an angle in a right triangle, then you often access the sine, the cosine or even the **tangent** back. The tangent of an angle for example corresponds to the length of his **Opposite cathete** divided by the length of its **Affiliate**.

If you now divide one length by the other, you will get one **Number as a result** and no angle. This number corresponds to the tangent of the angle under consideration. If you know the number and that **Determine the angle** want, you need them **Inverse function of the tangent**. And it is precisely this inverse function **Arctangent**. One often writes arctangent or abbreviates the function by arctan or arctan (x). Since the arctangent is the inverse function of the tangent, so is the notation common. However, it harbors the risk of being confused with the reciprocal of the tangent. The **So arctangent orders every number an angle** to. If you put this angle in the tangent function, you get the number again .

### Arc tangent as an inverse function

However, there is still one small difficulty to overcome. We want to draw your attention to the fact that the **Tangent function not injective **is. This means that one and the same function value is assumed several times. For example, the tangent of 45 ° is equal to one, as is the tangent of 405 °.

The **Tangent function** is namely **periodically** with a period of 180 °. You can see that by looking at their function graph.

Since the tangent function so **not injective** is, it is too **not bijective **and thus can **no reverse function** can be specified. Because it is not clear, for example, which angle the inverse function should assign to the number one. The 45 ° angle or the 405 ° angle? The tangent of both angles is the same. However, this problem can easily be avoided by using the **Restrict the tangent function to a range of 180 °**. The following interval is usually chosen:

or.

The function graph of the tangent clearly shows that in this area the tangent function is both injective and surjective and thus **bijective** is. The **Arctangent** so provides the **Inverse function of the tangent** which has been restricted to this area. The graph of the arctangent is obtained by mirroring the graph of the tangent function at the bisector.

The bisector corresponds to the graph of the function . Also for them **Cotangent function** there is only one **Inverse function**if you restrict it to a suitable interval. You limit it to the area or. one and its inverse function is called **Arcus cotangent**.

### Properties of the arctan

On the function graph of the **arctan** the following properties of the function can then be recognized:

Arctangent | |
---|---|

Definition set | the whole real numbers |

Image set | ]-90°,90°[ |

monotony | strictly monotonically increasing |

Symmetries | odd: arctan (-x) = - arctan (x) |

Asymptotes | -90 ° and 90 ° |

zeropoint | x = 0 |

Jump points | no |

Poles | no |

Extremes | no |

Turning point | (0,0) |

### Important functional values of the arctangent

It is also useful if you know common function values. Here are a few of them summarized.

### Approximately calculate the arctan

The **Arctangent** can **approximately** using the following formula **calculated** become:

### Series development

If you develop the arctangent in a **Taylor series**, the following expression is obtained.

### Derivative arctan

Generally applies to the **Derivation of the inverse function** a function The following:

So you get for the **arctan derivative** so:

The derivation of the arctangent can also be represented in another form:

### Indefinite integral

Furthermore it can be shown that the arctangent has the following antiderivative:

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