gives the arc cosine of the complex number .
- 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.
Examplesopen allclose all
Basic Examples (6)
Numerical Evaluation (6)
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)
Plot the ArcCos function:
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:
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:
Series Expansions (4)
Function Identities and Simplifications (3)
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:
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:
ArcCos is automatically returned as a special case for various mathematical functions:
Possible Issues (4)
Wolfram Research (1988), ArcCos, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCos.html (updated 2021).
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. Retrieved from https://reference.wolfram.com/language/ref/ArcCos.html