ArcTanDegrees

ArcTanDegrees[z]

gives the arc tangent in degrees of the complex number .

Details

  • ArcTanDegrees, along with other inverse trigonometric and trigonometric functions, is studied in high-school geometry courses and is also used in many scientific disciplines.
  • All results are given in degrees.
  • For real , the results are always in the range to .
  • ArcTanDegrees[z] returns the angle in degrees for which the ratio of the opposite side to the adjacent side of a right triangle is .
  • For certain special arguments, ArcTanDegrees automatically evaluates to exact values.
  • ArcTanDegrees can be evaluated to arbitrary numerical precision.
  • ArcTanDegrees automatically threads over lists.
  • ArcTanDegrees[z] has branch cut discontinuities in the complex plane running from to and to .
  • ArcTanDegrees can be used with Interval, CenteredInterval and Around objects.
  • Mathematical function, suitable for both symbolic and numerical manipulation.

Examples

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Basic Examples  (7)

Results are in degrees:

Calculate the angle BAC of this right triangle:

Calculate by hand:

The numerical value of this angle:

Solve an inverse trigonometric equation:

Solve an inverse trigonometric inequality:

Apply ArcTanDegrees to the following list:

Plot over a subset of the reals:

Series expansion at 0:

Scope  (40)

Numerical Evaluation  (6)

Evaluate numerically:

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate ArcTanDegrees 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 ArcTanDegrees function using MatrixFunction:

Specific Values  (5)

Values of ArcTanDegrees at fixed points:

Simple exact values are generated automatically:

Values at infinity:

Zero of ArcTanDegrees:

Find the value of satisfying equation :

Substitute in the value:

Visualize the result:

Visualization  (4)

Plot the ArcTanDegrees function:

Plot over a subset of the complexes:

Plot the real part of ArcTanDegrees:

Plot the imaginary part of ArcTanDegrees:

Polar plot with ArcTanDegrees:

Function Properties  (12)

ArcTanDegrees is defined for all real values:

Complex domain:

ArcTanDegrees achieves all real values from the interval :

The range for complex values:

ArcTanDegrees is an odd function:

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

ArcTanDegrees is an analytic function of over the reals:

It is neither analytic nor meromorphic over the complex plane:

ArcTanDegrees is an increasing function:

ArcTanDegrees is injective:

ArcTanDegrees is not surjective:

ArcTanDegrees is neither non-negative nor non-positive:

ArcTanDegrees has no singularities or discontinuities:

ArcTanDegrees is neither convex nor concave:

ArcSind is convex for x in [-10,0]:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (2)

Indefinite integral of ArcTanDegrees:

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

Series Expansions  (5)

Taylor expansion for ArcTanDegrees:

Plot the first three approximations for ArcTanDegrees around :

Asymptotic expansions at Infinity:

Asymptotic expansion at one of the singular points:

Find series expansions at branch points and branch cuts:

ArcTanDegrees can be applied to a power series:

Function Identities and Simplifications  (2)

Use FullSimplify to simplify expressions with ArcTanDegrees:

Use TrigToExp to express ArcTanDegrees using Log:

Function Representations  (1)

Represent using ArcCotDegrees:

Applications  (7)

Solve inverse trigonometric equations:

Solve an inverse trigonometric equation with a parameter:

Use Reduce to solve inequalities involving ArcTanDegrees:

Numerically find a root of a transcendental equation:

Plot the function to check if the solution is correct:

Plot the real and imaginary parts of ArcTanDegrees:

Different combinations of ArcTanDegrees with trigonometric functions:

Addition theorem for tangent function:

Properties & Relations  (5)

Compositions with the inverse trigonometric functions:

Use PowerExpand to disregard multivaluedness of the ArcTanDegrees:

Alternatively, evaluate under additional assumptions:

Branch cuts of ArcTanDegrees run along the imaginary axis:

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

FunctionExpand applied to ArcTanDegrees generates expressions in trigonometric functions in radians:

ExpToTrig applied to the outputs of TrigToExp will generate trigonometric functions in radians:

Possible Issues  (1)

Generically :

This differs from the original argument by a factor of :

Neat Examples  (2)

Solve trigonometric equations involving ArcTanDegrees:

Numerical value of this angle in degrees:

Plot ArcTanDegrees at integer points:

Wolfram Research (2024), ArcTanDegrees, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcTanDegrees.html.

Text

Wolfram Research (2024), ArcTanDegrees, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcTanDegrees.html.

CMS

Wolfram Language. 2024. "ArcTanDegrees." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ArcTanDegrees.html.

APA

Wolfram Language. (2024). ArcTanDegrees. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcTanDegrees.html

BibTeX

@misc{reference.wolfram_2025_arctandegrees, author="Wolfram Research", title="{ArcTanDegrees}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/ArcTanDegrees.html}", note=[Accessed: 30-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_arctandegrees, organization={Wolfram Research}, title={ArcTanDegrees}, year={2024}, url={https://reference.wolfram.com/language/ref/ArcTanDegrees.html}, note=[Accessed: 30-June-2025 ]}