ArcCosDegrees

ArcCosDegrees[z]

gives the arc cosine in degrees of the complex number .

Details

  • ArcCosDegrees, 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 between and , the results are always in the range to .
  • ArcCosDegrees[z] returns the angle in degrees for which the ratio of the adjacent side to the hypotenuse of a right triangle is .
  • For certain special arguments, ArcCosDegrees automatically evaluates to exact values.
  • ArcCosDegrees can be evaluated to arbitrary numerical precision.
  • ArcCosDegrees automatically threads over lists.
  • ArcCosDegrees[z] has branch cut discontinuities in the complex plane running from to and to .
  • ArcCosDegrees 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 ArcCosDegrees to the following list:

Plot over a subset of the reals:

Series expansion at 0:

Scope  (38)

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 ArcCosDegrees 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 ArcCosDegrees function using MatrixFunction:

Specific Values  (5)

Values of ArcCosDegrees at fixed points:

Simple exact values are generated automatically:

Values at infinity:

Zero of ArcCosDegrees:

Find the value of satisfying equation :

Substitute in the value:

Visualize the result:

Visualization  (4)

Plot the ArcCosDegrees function:

Plot over a subset of the complexes:

Plot the real part of ArcCosDegrees:

Plot the imaginary part of ArcCosDegrees:

Polar plot with ArcCosDegrees:

Function Properties  (10)

ArcCosDegrees is defined for all real values from the interval :

Complex domain is the whole plane:

ArcCosDegrees achieves all real values from the interval :

The range for complex values:

ArcCosDegrees is not an analytic function:

Nor is it meromorphic:

ArcCosDegrees is neither non-decreasing nor non-increasing:

It is monotonic over its real domain:

ArcCosDegrees is injective:

ArcCosDegrees is not surjective:

ArcCosDegrees is non-negative over its real domain:

ArcCosDegrees has both singularity and discontinuity in (-∞,-1] and [1,∞):

ArcCosDegrees is neither convex nor concave:

ArcCosDegrees is convex for x in [-1,0]:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (2)

Indefinite integral of ArcCosDegrees:

Definite integral of ArcCosDegrees over the entire real domain:

Series Expansions  (5)

Find the Taylor expansion using Series:

Plot the first three approximations for ArcCosDegrees around :

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

Find the series expansion at branch points and branch cuts:

ArcCosDegrees can be applied to power series:

Function Identities and Simplifications  (2)

Simplify expressions involving ArcCosDegrees:

Use TrigToExp to express through logarithms and square roots:

Function Representations  (1)

Represent using ArcSecDegrees:

Applications  (8)

Solve an inverse trigonometric equation:

Solve an inverse trigonometric equation with a parameter:

Get the zeros of ArcCosDegrees:

Use Reduce to solve inequalities involving ArcCosDegrees:

Numerically find a root of a transcendental equation:

Plot the function to check if the solution is correct:

Plot the real and imaginary part of ArcCosDegrees:

Plot the Riemann surface of ArcCosDegrees:

Find the angle between two vectors:

Properties & Relations  (5)

Compose with the inverse function:

Use PowerExpand to disregard multivaluedness of the ArcCosDegrees:

Alternatively, evaluate under additional assumptions:

This shows the branch cuts of the ArcCosDegrees function:

ArcCosDegrees gives the angle in degrees, while ArcCos gives the same angle in radians:

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

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

Possible Issues  (3)

Generically :

On branch cuts, machine-precision inputs can give numerically wrong answers:

The precision of the output can be much less than the precision of the input:

Neat Examples  (3)

Solve trigonometric equations involving ArcCosDegrees:

Numerical value of this angle in degrees:

Plot a specific ArcCosDegrees:

Plot ArcCosDegrees at integer points:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_arccosdegrees, author="Wolfram Research", title="{ArcCosDegrees}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/ArcCosDegrees.html}", note=[Accessed: 18-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_arccosdegrees, organization={Wolfram Research}, title={ArcCosDegrees}, year={2024}, url={https://reference.wolfram.com/language/ref/ArcCosDegrees.html}, note=[Accessed: 18-November-2024 ]}