ArcSinDegrees

ArcSinDegrees[z]

gives the arc sine in degrees of the complex number .

Details

ArcSinDegrees, along with other inverse trigonometric and trigonometric functions, is studied in high-school geometry courses and is also used in many scientific disciplines.
All results are given in degrees.
For real between and , the results are always in the range to .
ArcSinDegrees[z] returns the angle in degrees for which the ratio of the opposite side to the hypotenuse of a right triangle is .

For certain special arguments, ArcSinDegrees automatically evaluates to exact values.
ArcSinDegrees can be evaluated to arbitrary numerical precision.
ArcSinDegrees automatically threads over lists.
ArcSinDegrees[z] has branch cut discontinuities in the complex plane running from to and to .
ArcSinDegrees can be used with Interval, CenteredInterval and Around objects.
Mathematical function, suitable for both symbolic and numerical manipulation.

Examples

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Basic Examples (7)

Results are in degrees:

Calculate the angle ABC of this right triangle:

Calculate by hand:

The numerical value of this angle:

Solve an inverse trigonometric equation:

Solve an inverse trigonometric inequality:

Apply ArcSinDegrees to the following list:

Plot over a subset of the reals:

Series expansion at 0:

Scope (39)

Numerical Evaluation (6)

Evaluate numerically:

Evaluate to high precision:

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

ArcSinDegrees can take complex number inputs:

Evaluate ArcSinDegrees efficiently at high precision:

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

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix ArcSinDegrees function using MatrixFunction:

Specific Values (5)

Values of ArcSinDegrees at fixed points:

Simple exact values are generated automatically:

Values at infinity:

Zero of ArcSinDegrees:

Find the value of satisfying equation :

Substitute in the value:

Visualize the result:

Visualization (4)

Plot the ArcSinDegrees function:

Plot over a subset of the complexes:

Plot the real part of ArcSinDegrees:

Plot the imaginary part of ArcSinDegrees:

Polar plot with ArcSinDegrees:

Function Properties (11)

ArcSinDegrees is defined for all real values from the interval :

Complex domain is the whole plane:

ArcSinDegrees achieves all real values from the interval :

The range for complex values:

ArcSinDegrees is an odd function:

ArcSinDegrees is not an analytic function:

Nor is it meromorphic:

ArcSinDegrees is neither non-decreasing nor non-increasing:

It is monotonic over its real domain:

ArcSinDegrees is injective:

ArcSinDegrees is not surjective:

ArcSinDegrees is neither non-negative nor non-positive:

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

ArcSinDegrees is neither convex nor concave:

ArcSinDegrees is concave for x in [-1,0]:

TraditionalForm formatting:

Differentiation (3)

First derivative:

Higher derivatives:

Formula for the derivative:

Integration (2)

Indefinite integral of ArcSinDegrees:

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

Series Expansions (5)

Find the Taylor expansion using Series:

Plot the first three approximations for ArcSinDegrees around :

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

Find series expansions at branch points and branch cuts:

ArcSinDegrees can be applied to power series:

Function Identities and Simplifications (2)

Simplify expressions involving ArcSinDegrees:

Use TrigToExp to express through logarithms and square roots:

Function Representations (1)

Represent using ArcCscDegrees:

Applications (9)

Solve an inverse trigonometric equation:

Solve an inverse trigonometric equation with a parameter:

Get the zeros of ArcSinDegrees:

Use Reduce to solve inequalities involving ArcSinDegrees:

Numerically find a root of a transcendental equation:

Plot the function to check if the solution is correct:

Plot the real and imaginary parts of ArcSinDegrees:

Plot the Riemann surface of ArcSinDegrees:

Find the angle between two 3D vectors:

Different combinations of ArcSinDegrees with trigonometric functions:

Properties & Relations (5)

Compositions with the inverse trigonometric functions:

Use PowerExpand to disregard multivaluedness of the ArcSinDegrees:

Alternatively, evaluate under additional assumptions:

This shows the branch cuts of the ArcSinDegrees function:

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

FunctionExpand applied to ArcSinDegrees generates expressions in trigonometric functions in radians:

ExpToTrig applied to the outputs of TrigToExp will generate trigonometric functions in radians:

Possible Issues (3)

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:

Neat Examples (3)

Solve trigonometric equations involving ArcSinDegrees:

Numerical value of this angle in degrees:

Calculate numerical values of ArcSinDegrees by iteration:

Plot ArcSinDegrees at integer points:

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ArcSinDegrees

Details

Examples

Basic Examples (7)