Built-in Mathematica Symbol

ArcSin

ArcSin[z]
gives the arc sine

of the complex number

.

MORE INFORMATION

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 .

EXAMPLES

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Basic Examples (3)

Results are in radians:

Scope (7)

Evaluate numerically:

Evaluate for complex arguments:

Evaluate to high precision:

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

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

Simple exact values are generated automatically:

Parity transformation is automatically applied:

ArcSin threads element-wise over lists and matrices:

TraditionalForm formatting:

Generalizations & Extensions (5)

ArcSin can deal with real-valued intervals from

:

Infinite arguments give symbolic results:

ArcSin can be applied to power series:

Find series expansions at branch points and branch cuts:

ArcSin threads over explicit lists as well as over sparse arrays:

Applications (4)

Plot the real and imaginary parts of ArcSin:

Plot the Riemann surface of ArcSin:

Find the angle between two 3D vectors:

Modeling Levy's second arc sine law:

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:

Expand assuming real variables:

Solve an inverse trigonometric equation:

Solve for zeros:

Integrals:

Laplace transforms:

ArcSin is a special case of various mathematical functions:

Possible Issues (3)

Generically

:

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

In traditional form parentheses are needed around the argument:

Neat Examples (3)

Nested integrals:

Calculate numerical values by iteration:

Plot ArcSin at integer points: