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ArcCos
✖
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)Summary of the most common use cases

Results are in radians:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-u0d
Out[1]=1

Divide by Degree to get results in degrees:

In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-qw7
Out[2]=2

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-yz6
Out[1]=1

Plot over a subset of the complexes:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-kiedlx
Out[1]=1

Series expansion at 0:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-bq1
Out[1]=1

Asymptotic expansion at Infinity:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-klydni
Out[1]=1

Asymptotic expansion at a singular point:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-k12ao4
Out[1]=1

Scope  (41)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-pbj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-h5l
Out[2]=2

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-xkb
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-qkm
Out[2]=2

Evaluate for complex arguments:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-cj0
Out[1]=1

Evaluate ArcCos efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-di5gcr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-bq2c6r
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-c0rerv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-lmyeh7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0e7ph9ly-pirpwb
Out[3]=3

Or compute average-case statistical intervals using Around:

In[4]:=4
✖
https://wolfram.com/xid/0e7ph9ly-cw18bq
Out[4]=4

Compute the elementwise values of an array:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-thgd2
Out[1]=1

Or compute the matrix ArcCos function using MatrixFunction:

In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-o5jpo
Out[2]=2

Specific Values  (4)

Values of ArcCos at fixed points:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-nww7l
Out[1]=1

Values at infinity:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-bdij6w
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-drqkdo
Out[2]=2

Zero of ArcCos:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-cw39qs
Out[1]=1

Find the value of satisfying equation :

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-f2hrld
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-op0v0e
Out[2]=2

Substitute in the value:

In[3]:=3
✖
https://wolfram.com/xid/0e7ph9ly-yei1qb
Out[3]=3

Visualize the result:

In[4]:=4
✖
https://wolfram.com/xid/0e7ph9ly-d6prj
Out[4]=4

Visualization  (3)

Plot the ArcCos function:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-ecj8m7
Out[1]=1

Plot the real part of :

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-bo5grg
Out[1]=1

Plot the imaginary part of :

In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-d82jgn
Out[2]=2

Polar plot with :

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-epb4bn
Out[1]=1

Function Properties  (10)

ArcCos is defined for all real values from the interval :

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-cl7ele
Out[1]=1

Complex domain is the whole plane:

In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-de3irc
Out[2]=2

ArcCos achieves all real values from the interval :

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-evf2yr
Out[1]=1

Function range for arguments from the complex domain:

In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-fphbrc
Out[2]=2

ArcCos is not an analytic function:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-h5x4l2
Out[1]=1

Nor is it meromorphic:

In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-e434t9
Out[2]=2

ArcCos is neither non-decreasing nor non-increasing:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-g6kynf
Out[1]=1

It is monotonic over its real domain:

In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-nlz7s
Out[2]=2

ArcCos is injective:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-gi38d7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-ctca0g
Out[2]=2

ArcCos is not surjective:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-hkqec4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-hdm869
Out[2]=2

ArcCos is non-negative over its real domain:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-84dui
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-mdtl3h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-mn5jws
Out[2]=2

ArcCos is neither convex nor concave:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-kdss3
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-3qsfn

Differentiation  (3)

First derivative:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-mmas49
Out[1]=1

Higher derivatives:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-nfbe0l
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-fxwmfc
Out[2]=2

Formula for the ^(th) derivative:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-odmgl1
Out[1]=1

Integration  (3)

Indefinite integral of ArcCos:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-bponid
Out[1]=1

Definite integral of ArcCos over the entire real domain:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-b9jw7l
Out[1]=1

More integrals:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-fbqp9h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-pe0f3
Out[2]=2

Series Expansions  (4)

Find the Taylor expansion using Series:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-ewr1h8
Out[1]=1

Plot the first three approximations for ArcCos around :

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-binhar
Out[2]=2

General term in the series expansion of ArcCos:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-hx3l7
Out[1]=1

Find the series expansion at branch points and branch cuts:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-eib4fg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-zm631
Out[2]=2

ArcCos can be applied to power series:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-fbjcan
Out[1]=1

Function Identities and Simplifications  (3)

Simplify expressions involving ArcCos:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-gji1fc
Out[1]=1

Use TrigToExp to express through logarithms and square roots:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-nsm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-bqfsv5
Out[2]=2

Expand assuming real variables and :

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-z2g1m
Out[1]=1

Function Representations  (5)

Represent using ArcSec:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-bujw03
Out[1]=1

Representation through inverse Jacobi functions:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-m9whnz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-i0p6v1
Out[2]=2

Represent using Hypergeometric2F1:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-30gtga
Out[1]=1

Representation in terms of MeijerG:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-okel2r
Out[1]=1

ArcCos can be represented as a DifferentialRoot:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-bgjnbg
Out[1]=1

Applications  (5)Sample problems that can be solved with this function

Plot the real and imaginary part of ArcCos:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-wnb
Out[1]=1

Plot the Riemann surface of ArcCos:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-moo
Out[1]=1

Find the angle between two vectors:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-whr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-ban59i
Out[2]=2

Solve a differential equation:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-egz7aw
Out[1]=1

3D version of a phyllotaxis pattern:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-z28l1b
Out[1]=1

Properties & Relations  (9)Properties of the function, and connections to other functions

Compose with the inverse function:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-vkl
Out[1]=1

Use PowerExpand to disregard multivaluedness of the ArcCos:

In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-3u
Out[2]=2

Alternatively, evaluate under additional assumptions:

In[3]:=3
✖
https://wolfram.com/xid/0e7ph9ly-gywjjy
Out[3]=3

Use TrigToExp to express ArcCos through logarithms and square roots:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-v7c
Out[1]=1

This shows the branch cuts of the ArcCos function:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-sg7
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-0u4ktt
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-8h1c0s
Out[2]=2

Expand assuming real variables:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-se6
Out[1]=1

Solve an inverse trigonometric equation:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-mzm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-py7
Out[2]=2

Solve for zeros:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-l11
Out[1]=1

Laplace transforms:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-lvl
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-lf0
Out[1]=1

Possible Issues  (4)Common pitfalls and unexpected behavior

Generically :

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-gla
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-s47
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-k2i
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-oek
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-t4c
Out[1]=1

In traditional form, parentheses are needed around the argument:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-ppr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ph9ly-eqj
Out[2]=2

Neat Examples  (2)Surprising or curious use cases

Nested integrals:

In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-q11
Out[1]=1
In[1]:=1
✖
https://wolfram.com/xid/0e7ph9ly-eve8r7
Out[1]=1
Wolfram Research (1988), ArcCos, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCos.html (updated 2021).
✖
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).

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_arccos, author="Wolfram Research", title="{ArcCos}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/ArcCos.html}", note=[Accessed: 12-May-2025 ]}

✖
@misc{reference.wolfram_2025_arccos, author="Wolfram Research", title="{ArcCos}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/ArcCos.html}", note=[Accessed: 12-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_arccos, organization={Wolfram Research}, title={ArcCos}, year={2021}, url={https://reference.wolfram.com/language/ref/ArcCos.html}, note=[Accessed: 12-May-2025 ]}

✖
@online{reference.wolfram_2025_arccos, organization={Wolfram Research}, title={ArcCos}, year={2021}, url={https://reference.wolfram.com/language/ref/ArcCos.html}, note=[Accessed: 12-May-2025 ]}