Arctangent

Definition and basic equations

In trigonometry, the arctangent, also known as the inverse tangent, is a trigonometric function. It can be defined as an inverse of the tangent function

$\tan (x)=\frac{\sin (x)}{\cos (x)},$

where $x$ is an angle (argument), such that if $y=\tan (x)$, then the arctangent function

$\arctan (y)=x.$

This equation can be written in alternative notation as

${\tan }^{-1}(y)=x$

that is also widely used in mathematics.

Generally, the argument may be a complex number $z=x+iy$ , where $i=\sqrt{-1}$ is known as the imaginary unit and for $z\in \mathbb{ℂ}$ the arctangent function can be represented in form of the Maclaurin series expansion

$\arctan (z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{z}^{2n+1}}{2n+1}.$

Euler's formula for the arctangent function is a rapidly convergent series expansion, given by

$\arctan (x)=\sum _{n=0}^{\mathrm{\infty }}\frac{2^{2n}(n!{)}^2}{2n+1}\frac{x^{2n+1}}{(1+x^2{)}^{n+1}},$

where $x\in \mathbb{ℝ}$. Math&App (talk) 15:35, 6 October 2020 (UTC)

References

This article "Arctangent" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Arctangent. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

---