ArcCos

ArcCos[z]

gives the arc cosine of the complex number .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • All results are given in radians.
  • For real between and , the results are always in the range to .
  • For certain special arguments, ArcCos automatically evaluates to exact values.
  • ArcCos can be evaluated to arbitrary numerical precision.
  • ArcCos automatically threads over lists.
  • ArcCos[z] has branch cut discontinuities in the complex plane running from to and to .
  • ArcCos can be used with Interval and CenteredInterval objects. »

Background & Context

  • ArcCos is the inverse cosine function. For a real number , ArcCos[x] represents the radian angle measure such that .
  • ArcCos automatically threads over lists. For certain special arguments, ArcCos automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcCos may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcCos include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
  • ArcCos is defined for complex argument via . ArcCos[z] has branch cut discontinuities in the complex plane.
  • Related mathematical functions include Cos, ArcSin, and ArcCosh.

Examples

open allclose all

Basic Examples  (6)

Results are in radians:

Divide by Degree to get results in degrees:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at 0:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

Scope  (41)

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 ArcCos efficiently at high precision:

ArcCos threads elementwise over lists and matrices:

ArcCos can be used with Interval and CenteredInterval objects:

Specific Values  (4)

Values of ArcCos at fixed points:

Values at infinity:

Zero of ArcCos:

Find the value of satisfying equation :

Substitute in the value:

Visualize the result:

Visualization  (3)

Plot the ArcCos function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

Function Properties  (10)

ArcCos is defined for all real values from the interval :

Complex domain is the whole plane:

ArcCos achieves all real values from the interval :

Function range for arguments from the complex domain:

ArcCos is not an analytic function:

Nor is it meromorphic:

ArcCos is neither non-decreasing nor non-increasing:

It is monotonic over its real domain:

ArcCos is injective:

ArcCos is not surjective:

ArcCos is non-negative over its real domain:

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

ArcCos is neither convex nor concave:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of ArcCos:

Definite integral of ArcCos over the entire real domain:

More integrals:

Series Expansions  (4)

Find the Taylor expansion using Series:

Plot the first three approximations for ArcCos around :

General term in the series expansion of ArcCos:

Find the series expansion at branch points and branch cuts:

ArcCos can be applied to power series:

Function Identities and Simplifications  (3)

Simplify expressions involving ArcCos:

Use TrigToExp to express through logarithms and square roots:

Expand assuming real variables and :

Function Representations  (5)

Represent using ArcSec:

Representation through inverse Jacobi functions:

Represent using Hypergeometric2F1:

Representation in terms of MeijerG:

ArcCos can be represented as a DifferentialRoot:

Applications  (5)

Plot the real and imaginary part of ArcCos:

Plot the Riemann surface of ArcCos:

Find the angle between two vectors:

Solve a differential equation:

3D version of a phyllotaxis pattern:

Properties & Relations  (8)

Compose with the inverse function:

Use PowerExpand to disregard multivaluedness of the ArcCos:

Alternatively, evaluate under additional assumptions:

Use TrigToExp to express ArcCos through logarithms and square roots:

This shows the branch cuts of the ArcCos function:

Expand assuming real variables:

Solve an inverse trigonometric equation:

Solve for zeros:

Laplace transforms:

ArcCos is automatically returned as a special case for various mathematical functions:

Possible Issues  (4)

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:

In traditional form, parentheses are needed around the argument:

Neat Examples  (2)

Nested integrals:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2023_arccos, author="Wolfram Research", title="{ArcCos}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/ArcCos.html}", note=[Accessed: 01-October-2023 ]}

BibLaTeX

@online{reference.wolfram_2023_arccos, organization={Wolfram Research}, title={ArcCos}, year={2021}, url={https://reference.wolfram.com/language/ref/ArcCos.html}, note=[Accessed: 01-October-2023 ]}