Arctan equals inverse tan
In this article, we'll do theArctangentand deal with it in detail. Among other things, we explain theproperties of the arctangent and name its most importantFunction values. We'll also show you how to do this toocalculate approximately and the function in a Develop series can. Finally we'll show you howAntiderivative 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:
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 functionif 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:
|Definition set||the whole real numbers|
|monotony||strictly monotonically increasing|
|Symmetries||odd: arctan (-x) = - arctan (x)|
|Asymptotes||-90 ° and 90 °|
|zeropoint||x = 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:
If you develop the arctangent in a Taylor series, the following expression is obtained.
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:
Furthermore it can be shown that the arctangent has the following antiderivative:
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