ArcTan
ArcTan[z]
gives the arc tangent of the complex number
.
ArcTan[x,y]
gives the arc tangent of , taking into account which quadrant the point
is in.
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- All results are given in radians.
- For real
, the results are always in the range
to
.
- For certain special arguments, ArcTan automatically evaluates to exact values.
- ArcTan can be evaluated to arbitrary numerical precision.
- ArcTan automatically threads over lists.
- ArcTan[z] has branch cut discontinuities in the complex
plane running from
to
and
to
.
- If
or
is complex, then ArcTan[x,y] gives
. When
, ArcTan[x,y] gives the number
such that
and
.
Background & Context
- ArcTan is the inverse tangent function. For a real number x, ArcTan[x] represents the radian angle measure
such that
. The two-argument form ArcTan[x,y] represents the arc tangent of y/x, taking into account the quadrant in which the point
lies. It therefore gives the angular position (expressed in radians) of the point measured from the positive
axis. ArcTan is consequently useful when converting from Cartesian to polar coordinate systems and for finding the phase
in phasor notation
.
- ArcTan automatically threads over lists. For certain special exact arguments, ArcTan automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcTan may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcTan include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
- ArcTan is defined for complex argument
via
. ArcTan[z] has branch cut discontinuities in the complex
plane.
- Related mathematical functions include Arg, Tan, ArcCot, ArcTanh, and Gudermannian.
Examples
open allclose allBasic Examples (7)
Scope (42)
Numerical Evaluation (6)
Evaluate using the two-argument form:
The precision of the output tracks the precision of the input:
Evaluate for complex arguments:
The two-argument form supports complex numbers:
Evaluate ArcTan efficiently at high precision:
ArcTan can deal with real-valued intervals:
ArcTan threads elementwise over lists and matrices:
Specific Values (6)
Visualization (4)
Function Properties (5)
ArcTan is defined for all real values:
ArcTan achieves all real values from the interval :
Function range for arguments from the complex domain:
ArcTan is an odd function:
ArcTan has the mirror property :
TraditionalForm formatting:
Integration (3)
Series Expansions (4)
Integral Transforms (3)
Function Identities and Simplifications (3)
Function Representations (5)
Represent using ArcCot:
Representation through inverse Jacobi functions:
Represent using Hypergeometric2F1:
ArcTan can be represented in terms of MeijerG:
ArcTan can be represented as a DifferentialRoot:
Applications (5)
Properties & Relations (4)
Possible Issues (1)
Because ArcTan is a multivalued function,
Text
Wolfram Research (1988), ArcTan, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcTan.html.
BibTeX
BibLaTeX
CMS
Wolfram Language. 1988. "ArcTan." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ArcTan.html.
APA
Wolfram Language. (1988). ArcTan. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcTan.html