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ArcTan

gives the arc tangent of the complex number .

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

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Basic Examples (7)

Results are in radians:

Divide by Degree to get results in degrees:

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

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at 0:

Asymptotic expansions at Infinity:

Asymptotic expansion at one of the singular points:

Scope (49)

Numerical Evaluation (6)

Evaluate numerically:

Evaluate to high precision:

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:

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

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix ArcTan function using MatrixFunction:

Specific Values (6)

Values of ArcTan at fixed points:

The angles of all points with integer coordinates between and :

Values at infinity:

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

Zero of ArcTan:

Find the value of satisfying equation :

Substitute in the value:

Visualize the result:

Visualization (4)

Plot the ArcTan function:

Plot the two-argument ArcTan function in the plane:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

Function Properties (12)

ArcTan is defined for all real values:

Complex domain:

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 $tan^(-1)(TemplateBox[{x}, Conjugate])=TemplateBox[{{{tan, ^, {(, {-, 1}, )}}, (, x, )}}, Conjugate]$ :

is an analytic function of over the reals:

It is neither analytic nor meromorphic over the complex plane:

is not analytic over the reals:

is an increasing function:

ArcTan is injective:

ArcTan is not surjective:

ArcTan is neither non-negative nor non-positive:

has no singularities or discontinuities:

is singular when :

ArcTan is neither convex nor concave:

TraditionalForm formatting:

Differentiation (3)

First derivative:

Higher derivatives:

Formula for the derivative:

Integration (3)

Indefinite integral of ArcTan:

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

More integrals:

Series Expansions (4)

Taylor expansion for ArcTan:

Plot the first three approximations for ArcTan around :

General term in the series expansion of ArcTan:

Find series expansions at branch points and branch cuts:

ArcTan can be applied to a power series:

Integral Transforms (3)

Compute the Laplace transform using LaplaceTransform:

InverseFourierTransform:

MellinTransform:

Function Identities and Simplifications (3)

Simplify expressions involving ArcTan:

Use TrigToExp to express ArcTan using Log:

Expand assuming real variables and :

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 (9)

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

They total to 90°:

Find integrals of rational functions in terms of ArcTan:

Addition theorem for tangent function:

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

Solve a differential equation:

Branch cuts of ArcTan run along the imaginary axis:

Polar decomposition of a complex number:

Special solution of the sine–Gordon equation:

Check the solution:

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

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

Properties & Relations (5)

Use TrigToExp to express ArcTan using Log:

Use FullSimplify to simplify expressions with ArcTan:

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

ArcTan is a special case of some special functions:

Use Reduce to solve inequalities involving ArcTan:

Possible Issues (1)

Because ArcTan is a multivalued function,

This differs from the original argument by a factor of :

Neat Examples (1)

Expansion about the branch point :

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