WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

ArcTan
✖
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 .
  • ArcTan can be used with Interval and CenteredInterval objects. »

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 x+ⅈ y=TemplateBox[{z}, Abs]ⅇ^(ⅈ phi).
  • 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 all

Basic Examples  (7)Summary of the most common use cases

Results are in radians:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-xdt
Out[1]=1

Divide by Degree to get results in degrees:

In[3]:=3
✖
https://wolfram.com/xid/0tzy5776-qw7
Out[3]=3

ArcTan[x,y] gives the angle of the point {x,y}:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-v9x
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-jff
Out[2]=2

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-fkk
Out[1]=1

Plot over a subset of the complexes:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-kiedlx
Out[1]=1

Series expansion at 0:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-rur
Out[1]=1

Asymptotic expansions at Infinity:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-klydni
Out[1]=1

Asymptotic expansion at one of the singular points:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-k12ao4
Out[1]=1

Scope  (49)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-mf9f3x
Out[1]=1
In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-iey7b3
Out[1]=1

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-gz8fud
Out[1]=1

Evaluate using the two-argument form:

In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-c09w30
Out[2]=2

The precision of the output tracks the precision of the input:

In[3]:=3
✖
https://wolfram.com/xid/0tzy5776-gvz0ta
Out[3]=3

Evaluate for complex arguments:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-b7vho4
Out[1]=1

The two-argument form supports complex numbers:

In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-kuxe50
Out[2]=2

Evaluate ArcTan efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-di5gcr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-bq2c6r
Out[2]=2

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-nagbar
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-lmyeh7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tzy5776-blphx
Out[3]=3

Or compute average-case statistical intervals using Around:

In[4]:=4
✖
https://wolfram.com/xid/0tzy5776-cw18bq
Out[4]=4

Compute the elementwise values of an array:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-thgd2
Out[1]=1

Or compute the matrix ArcTan function using MatrixFunction:

In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-o5jpo
Out[2]=2

Specific Values  (6)

Values of ArcTan at fixed points:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-nww7l
Out[1]=1

The angles of all points with integer coordinates between and :

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-ro6wem

Values at infinity:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-ec62ur
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-ihfz5
Out[2]=2

Values at infinity of the ArcTan[x,y] form:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-f97fg2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-omp91
Out[2]=2

Zero of ArcTan:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-cw39qs
Out[1]=1

Find the value of satisfying equation :

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-f2hrld
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-op0v0e
Out[2]=2

Substitute in the value:

In[3]:=3
✖
https://wolfram.com/xid/0tzy5776-yei1qb
Out[3]=3

Visualize the result:

In[4]:=4
✖
https://wolfram.com/xid/0tzy5776-bxotei
Out[4]=4

Visualization  (4)

Plot the ArcTan function:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-ecj8m7
Out[1]=1

Plot the two-argument ArcTan function in the plane:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-4i0jx0
Out[1]=1

Plot the real part of :

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-bo5grg
Out[1]=1

Plot the imaginary part of :

In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-bkmztt
Out[2]=2

Polar plot with :

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-epb4bn
Out[1]=1

Function Properties  (12)

ArcTan is defined for all real values:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-cl7ele
Out[1]=1

Complex domain:

In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-de3irc
Out[2]=2

ArcTan achieves all real values from the interval :

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-evf2yr
Out[1]=1

Function range for arguments from the complex domain:

In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-fphbrc
Out[2]=2

ArcTan is an odd function:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-dnla5q
Out[1]=1

ArcTan has the mirror property tan^(-1)(TemplateBox[{x}, Conjugate])=TemplateBox[{{{tan, ^, {(, {-, 1}, )}}, (, x, )}}, Conjugate]:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-heoddu
Out[1]=1

is an analytic function of over the reals:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-h5x4l2
Out[1]=1

It is neither analytic nor meromorphic over the complex plane:

In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-nkbi2f
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0tzy5776-e434t9
Out[3]=3

is not analytic over the reals:

In[4]:=4
✖
https://wolfram.com/xid/0tzy5776-38qulz
Out[4]=4

is an increasing function:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-g6kynf
Out[1]=1

ArcTan is injective:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-gi38d7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-ctca0g
Out[2]=2

ArcTan is not surjective:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-hkqec4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-hdm869
Out[2]=2

ArcTan is neither non-negative nor non-positive:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-84dui
Out[1]=1

has no singularities or discontinuities:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-mdtl3h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-mn5jws
Out[2]=2

is singular when :

In[3]:=3
✖
https://wolfram.com/xid/0tzy5776-rwegqb
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0tzy5776-d9bjn5
Out[4]=4

ArcTan is neither convex nor concave:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-kdss3
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-j24ade
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-ys3b2

Differentiation  (3)

First derivative:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-mmas49
Out[1]=1

Higher derivatives:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-nfbe0l
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-fxwmfc
Out[2]=2

Formula for the ^(th) derivative:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-odmgl1
Out[1]=1

Integration  (3)

Indefinite integral of ArcTan:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-bponid
Out[1]=1

Definite integral of ArcTan over an interval centered at the origin is 0:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-b9jw7l
Out[1]=1

More integrals:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-gzsps8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-k7js0j
Out[2]=2

Series Expansions  (4)

Taylor expansion for ArcTan:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-ewr1h8
Out[1]=1

Plot the first three approximations for ArcTan around :

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-binhar
Out[2]=2

General term in the series expansion of ArcTan:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-cvfu4n
Out[1]=1

Find series expansions at branch points and branch cuts:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-okams
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-i50ap2
Out[2]=2

ArcTan can be applied to a power series:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-c4r5wa
Out[1]=1

Integral Transforms  (3)

Compute the Laplace transform using LaplaceTransform:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-tq4
Out[1]=1

InverseFourierTransform:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-dh37pf
Out[1]=1

MellinTransform:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-7mn4u
Out[1]=1

Function Identities and Simplifications  (3)

Simplify expressions involving ArcTan:

In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-gji1fc
Out[2]=2

Use TrigToExp to express ArcTan using Log:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-c0uhue
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-idssh
Out[2]=2

Expand assuming real variables and :

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-mcb
Out[1]=1

Function Representations  (5)

Represent using ArcCot:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-bujw03
Out[1]=1

Representation through inverse Jacobi functions:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-m9whnz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-i0p6v1
Out[2]=2

Represent using Hypergeometric2F1:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-gdj3ez
Out[1]=1

ArcTan can be represented in terms of MeijerG:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-ey4sde
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-ctzqg1
Out[2]=2

ArcTan can be represented as a DifferentialRoot:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-b6y0sk
Out[1]=1

Applications  (9)Sample problems that can be solved with this function

Find angles of the right triangle with sides 3, 4 and hypotenuse 5:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-fhryb9
Out[1]=1

They total to 90°:

In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-kypfha
Out[2]=2

Find integrals of rational functions in terms of ArcTan:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-ca671l
Out[1]=1

Addition theorem for tangent function:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-bqgplf
Out[1]=1

Find the slope angle of the line through a pair of points and :

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-gec4ik
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-cd0bnx
Out[2]=2

Solve a differential equation:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-erbom5
Out[1]=1

Branch cuts of ArcTan run along the imaginary axis:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-ds4pr
Out[1]=1

Polar decomposition of a complex number:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-k684rt
Out[1]=1

Special solution of the sine–Gordon equation:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-g76

Check the solution:

In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-xbl
Out[2]=2

The cumulative distribution function (CDF) of the standard distribution of the hyperbolic secant is given in terms of ArcTan:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-bnplmf

This is a scaled and shifted version of the Gudermannian function:

In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-6olgr
Out[2]=2

Properties & Relations  (5)Properties of the function, and connections to other functions

Use TrigToExp to express ArcTan using Log:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-ncs
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-qlcrf4
Out[2]=2

Use FullSimplify to simplify expressions with ArcTan:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-ogg
Out[1]=1

ArcTan gives the angle in radians, while ArcTanDegrees gives the same angle in degrees:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-0u4ktt
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-8h1c0s
Out[2]=2

ArcTan is a special case of some special functions:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-ys1mr
Out[1]=1

Use Reduce to solve inequalities involving ArcTan:

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-d4vbx8
Out[1]=1

Possible Issues  (1)Common pitfalls and unexpected behavior

Because ArcTan is a multivalued function,

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-gewucp
Out[1]=1

This differs from the original argument by a factor of :

In[2]:=2
✖
https://wolfram.com/xid/0tzy5776-5x8e3
Out[2]=2

Neat Examples  (1)Surprising or curious use cases

Expansion about the branch point :

In[1]:=1
✖
https://wolfram.com/xid/0tzy5776-o6c
Out[1]=1
Wolfram Research (1988), ArcTan, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcTan.html (updated 2021).
✖
Wolfram Research (1988), ArcTan, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcTan.html (updated 2021).

Text

Wolfram Research (1988), ArcTan, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcTan.html (updated 2021).

✖
Wolfram Research (1988), ArcTan, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcTan.html (updated 2021).

CMS

Wolfram Language. 1988. "ArcTan." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcTan.html.

✖
Wolfram Language. 1988. "ArcTan." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. 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

✖
Wolfram Language. (1988). ArcTan. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcTan.html

BibTeX

@misc{reference.wolfram_2025_arctan, author="Wolfram Research", title="{ArcTan}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/ArcTan.html}", note=[Accessed: 22-June-2025 ]}

✖
@misc{reference.wolfram_2025_arctan, author="Wolfram Research", title="{ArcTan}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/ArcTan.html}", note=[Accessed: 22-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_arctan, organization={Wolfram Research}, title={ArcTan}, year={2021}, url={https://reference.wolfram.com/language/ref/ArcTan.html}, note=[Accessed: 22-June-2025 ]}

✖
@online{reference.wolfram_2025_arctan, organization={Wolfram Research}, title={ArcTan}, year={2021}, url={https://reference.wolfram.com/language/ref/ArcTan.html}, note=[Accessed: 22-June-2025 ]}