gives the arc sine 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, ArcSin automatically evaluates to exact values.
- ArcSin can be evaluated to arbitrary numerical precision.
- ArcSin automatically threads over lists.
- ArcSin[z] has branch cut discontinuities in the complex plane running from to and to .
- ArcSin can be used with Interval and CenteredInterval objects. »
Background & Context
- ArcSin is the inverse sine function. For a real number , ArcSin[x] represents the radian angle measure such that .
- ArcSin automatically threads over lists. For certain special arguments, ArcSin automatically evaluates to exact values. When given exact numeric expressions as arguments, ArcSin may be evaluated to arbitrary numeric precision. Operations useful for manipulation of symbolic expressions involving ArcSin include FunctionExpand, TrigToExp, TrigExpand, Simplify, and FullSimplify.
- ArcSin is defined for complex argument via . ArcSin[z] has branch cut discontinuities in the complex plane.
- Related mathematical functions include Sin, ArcCos, InverseHaversine, and ArcSinh.
Examplesopen allclose all
Basic Examples (6)
Numerical Evaluation (6)
ArcSin can take complex number inputs:
Evaluate ArcSin efficiently at high precision:
ArcSin threads elementwise over lists and matrices:
Specific Values (4)
Plot the ArcSin function:
Function Properties (11)
ArcSin is defined for all real values from the interval :
ArcSin achieves all real values from the interval :
ArcSin is an odd function:
ArcSin is not an analytic function:
ArcSin is neither non-decreasing nor non-increasing:
ArcSin is injective:
ArcSin is not surjective:
ArcSin is neither non-negative nor non-positive:
ArcSin has both singularity and discontinuity in (-∞,-1] and [1,∞):
ArcSin is neither convex nor concave:
Series Expansions (4)
Function Identities and Simplifications (3)
Properties & Relations (8)
Use TrigToExp to express through logarithms and square roots:
This shows the branch cuts of the ArcSin function:
ArcSin is a special case of various mathematical functions:
Possible Issues (4)
Neat Examples (3)
Plot ArcSin at integer points:
Wolfram Research (1988), ArcSin, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcSin.html (updated 2021).
Wolfram Language. 1988. "ArcSin." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/ArcSin.html.
Wolfram Language. (1988). ArcSin. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcSin.html