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

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

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

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

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

References

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