ArcSin

ArcSin[z]

gives the arc sine 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, 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 $sin^(-1)(z)=-ⅈ log(sqrt(TemplateBox[{{1, -, {z, ^, 2}}}, Abs]) ⅇ^(1/2 ⅈ arg(1-z^2))+ⅈ z)$ . ArcSin[z] has branch cut discontinuities in the complex plane.
Related mathematical functions include Sin, ArcCos, InverseHaversine, and ArcSinh.

Examples

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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 (42)

Numerical Evaluation (6)

Evaluate numerically:

Evaluate to high precision:

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

ArcSin can take complex number inputs:

Evaluate ArcSin efficiently at high precision:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix ArcSin function using MatrixFunction:

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

Or compute average-case statistical intervals using Around:

Specific Values (4)

Values of ArcSin at fixed points:

Values at infinity:

Zero of ArcSin:

Find the value of satisfying equation :

Substitute in the value:

Visualize the result:

Visualization (3)

Plot the ArcSin function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

Function Properties (11)

ArcSin is defined for all real values from the interval :

Complex domain is the whole plane:

ArcSin achieves all real values from the interval :

Function range for arguments from the complex domain:

ArcSin is an odd function:

ArcSin is not an analytic function:

Nor is it meromorphic:

ArcSin is neither non-decreasing nor non-increasing:

It is monotonic over its real domain:

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:

TraditionalForm formatting:

Differentiation (3)

First derivative:

Higher derivatives:

Formula for the derivative:

Integration (3)

Indefinite integral of ArcSin:

Definite integral of ArcSin over an interval centered at the origin is 0:

More integrals:

Series Expansions (4)

Find the Taylor expansion using Series:

Plot the first three approximations for ArcSin around :

General term in the series expansion of ArcSin:

Find series expansions at branch points and branch cuts:

ArcSin can be applied to power series:

Function Identities and Simplifications (3)

Simplify expressions involving ArcSin:

Use TrigToExp to express through logarithms and square roots:

Expand assuming real variables and :

Function Representations (5)

Represent using ArcCsc:

Representation through inverse Jacobi functions:

Represent using Hypergeometric2F1:

Representation in terms of MeijerG:

ArcSin can be represented as a DifferentialRoot:

Applications (6)

Plot the real and imaginary parts of ArcSin:

Plot the Riemann surface of ArcSin:

Find the angle between two 3D vectors:

Modeling Lévy's second arc sine law:

Solve the ArcSin differential equation:

Conformal map from an ellipse to the unit disk:

Visualize the map:

Properties & Relations (9)

Compose with the inverse function:

Use PowerExpand to disregard multivaluedness of the ArcSin:

Alternatively, evaluate under additional assumptions:

Use TrigToExp to express through logarithms and square roots:

This shows the branch cuts of the ArcSin function:

ArcSin gives the angle in radians, while ArcSinDegrees gives the same angle in degrees:

Expand assuming real variables:

Solve an inverse trigonometric equation:

Solve for zeros:

Laplace transforms:

ArcSin is a special case of 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 lower than the precision of the input:

In traditional form, parentheses are needed around the argument:

Neat Examples (3)

Nested integrals:

Calculate numerical values by iteration:

Plot ArcSin at integer points:

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