ArcCotDegrees

ArcCotDegrees[z]

gives the arc cotangent in degrees of the complex number .

Details

  • ArcCotDegrees, 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 , excluding 0.
  • ArcCotDegrees[z] returns the angle in degrees for which the ratio of the adjacent side to the opposite side of a right triangle is .
  • For certain special arguments, ArcCotDegrees automatically evaluates to exact values.
  • ArcCotDegrees can be evaluated to arbitrary numerical precision.
  • ArcCotDegrees automatically threads over lists.
  • ArcCotDegrees[z] has a branch cut discontinuity in the complex plane running from to .
  • ArcCotDegrees 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 ArcCotDegrees to the following list:

Plot over a subset of the reals:

Asymptotic expansion at Infinity:

Scope  (39)

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 ArcCotDegrees 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 ArcCotDegrees function using MatrixFunction:

Specific Values  (5)

Values of ArcCotDegrees at fixed points:

Simple exact values are generated automatically:

Values at infinity:

Singular points of ArcCotDegrees:

Find the value of satisfying equation :

Substitute in the value:

Visualize the result:

Visualization  (4)

Plot the ArcCotDegrees function:

Plot over a subset of the complexes:

Plot the real part of ArcCotDegrees:

Plot the imaginary part of ArcCotDegrees:

Polar plot with ArcCotDegrees:

Function Properties  (12)

ArcCotDegrees is defined for all real values:

Complex domain:

ArcCotDegrees achieves all real values except 0 from the interval :

The range for complex values:

ArcCotDegrees is an odd function:

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

ArcCotDegrees is not an analytic function:

Nor is it meromorphic:

ArcCotDegrees is neither non-decreasing nor non-increasing:

ArcCotDegrees is injective:

ArcCotDegrees is not surjective:

ArcCotDegrees is neither non-negative nor non-positive:

ArcCotDegrees has both singularity and discontinuity at zero:

ArcCotDegrees is neither convex nor concave:

ArcCotDegrees is convex for x in [0,100]:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (2)

Indefinite integral of ArcCotDegrees:

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

Series Expansions  (4)

Find the Taylor expansion using Series:

Plot the first three approximations for ArcCotDegrees around :

Find series expansions at branch points and branch cuts:

Asymptotic expansion at a singular point:

ArcCotDegrees can be applied to a power series:

Function Identities and Simplifications  (2)

Use FullSimplify to simplify expressions with ArcCotDegrees:

Use TrigToExp to express ArcCotDegrees using Log:

Function Representations  (1)

Represent using ArcTanDegrees:

Applications  (8)

Solve inverse trigonometric equations:

Solve an inverse trigonometric equation with a parameter:

Use Reduce to solve inequalities involving ArcCotDegrees:

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 ArcCotDegrees:

Different combinations of ArcCotDegrees with trigonometric functions:

Addition theorem for cotangent function:

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

They total to 90°:

Properties & Relations  (5)

Compositions with the inverse trigonometric functions:

Use PowerExpand to disregard multivaluedness of the ArcCotDegrees:

Alternatively, evaluate under additional assumptions:

Branch cut of ArcCotDegrees runs along the imaginary axis:

ArcCotDegrees gives the angle in degrees, while ArcCot gives the same angle in radians:

FunctionExpand applied to ArcCotDegrees 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 ArcCotDegrees:

Numerical value of this angle in degrees:

Plot ArcCotDegrees at integer points:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_arccotdegrees, author="Wolfram Research", title="{ArcCotDegrees}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/ArcCotDegrees.html}", note=[Accessed: 21-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_arccotdegrees, organization={Wolfram Research}, title={ArcCotDegrees}, year={2024}, url={https://reference.wolfram.com/language/ref/ArcCotDegrees.html}, note=[Accessed: 21-November-2024 ]}