ArcSinDegrees
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
open all close allBasic Examples (7)
ArcSinDegrees[1]Calculate the angle ABC of this right triangle:
β = ArcSinDegrees[3 / 8]The numerical value of this angle:
%//NSolve an inverse trigonometric equation:
Solve[ArcSinDegrees[x] == 60, x]Solve an inverse trigonometric inequality:
Reduce[ArcSinDegrees[x] > 60, x]Apply ArcSinDegrees to the following list:
ArcSinDegrees[{0, 1 / 2, 1 / Sqrt[2], Sqrt[3] / 2, 1}]Plot over a subset of the reals:
Plot[ArcSinDegrees[x], {x, -1, 1}]Series[ArcSinDegrees[x], {x, 0, 10}]Scope (39)
Numerical Evaluation (6)
ArcSinDegrees[0.4]ArcSinDegrees[1.5]N[ArcSinDegrees[1 / 3], 50]The precision of the output tracks the precision of the input:
ArcSinDegrees[0.33333333333333333333333333333333333333333333]ArcSinDegrees can take complex number inputs:
ArcSinDegrees[2.5 + I]Evaluate ArcSinDegrees efficiently at high precision:
ArcSinDegrees[0.4`500]//TimingArcSinDegrees[0.4`100000];//TimingCompute worst-case guaranteed intervals using Interval and CenteredInterval objects:
ArcSinDegrees[Interval[{-1 / 3, 1 / 2}]]ArcSinDegrees[CenteredInterval[1 / 2, 1 / 100]]ArcSinDegrees[CenteredInterval[2 + 3I, (1 + I) / 100]]Or compute average-case statistical intervals using Around:
ArcSinDegrees[Around[.9, 0.01]]Compute the elementwise values of an array:
ArcSinDegrees[{{1, 1 / Sqrt[2]}, {0, 1 / Sqrt[2]}}]Or compute the matrix ArcSinDegrees function using MatrixFunction:
MatrixFunction[ArcSinDegrees[#]&, {{1, 1 / Sqrt[2]}, {0, 1 / Sqrt[2]}}]Specific Values (5)
Values of ArcSinDegrees at fixed points:
Table[ArcSinDegrees[(n/2)], {n, -2, 2}]Simple exact values are generated automatically:
ArcSinDegrees[(1/Sqrt[2])]ArcSinDegrees[Infinity]ArcSinDegrees[ComplexInfinity]Zero of ArcSinDegrees:
ArcSinDegrees[0]Find the value of 
satisfying equation 
:
f[x_] := ArcSinDegrees[x] - 60;sol = Solve[f[x] == 0, x]xzero = x /. First[sol]Plot[f[x], {x, -1, 1}, Rule[...]]Visualization (4)
Plot the ArcSinDegrees function:
Plot[ArcSinDegrees[x], {x, -1, 1}]Plot over a subset of the complexes:
ComplexPlot3D[ArcSinDegrees[z], {z, -4 - 2I, 4 + 2I}, Rule[...]]Plot the real part of ArcSinDegrees:
ComplexContourPlot[Re[ArcSinDegrees[z]], {z, -3π - 3I, 3π + 3I}, ...]Plot the imaginary part of ArcSinDegrees:
ComplexContourPlot[Im[ArcSinDegrees[z]], {z, -π - 3I, π + 3I}, Rule[...]]Polar plot with ArcSinDegrees:
Table[PolarPlot[ArcSinDegrees[k ϕ], {ϕ, -π, π}, ...], {k, 1, 4}]Function Properties (11)
ArcSinDegrees is defined for all real values from the interval 
:
FunctionDomain[ArcSinDegrees[x], x]Complex domain is the whole plane:
FunctionDomain[ArcSinDegrees[z], z, Complexes]ArcSinDegrees achieves all real values from the interval 
:
FunctionRange[ArcSinDegrees[x], x, y]FunctionRange[ArcSinDegrees[x], x, y, Complexes]ArcSinDegrees is an odd function:
ArcSinDegrees[-x]ArcSinDegrees is not an analytic function:
FunctionAnalytic[ArcSinDegrees[x], x]FunctionMeromorphic[ArcSinDegrees[x], x]ArcSinDegrees is neither non-decreasing nor non-increasing:
FunctionMonotonicity[ArcSinDegrees[x], x]It is monotonic over its real domain:
FunctionMonotonicity[{ArcSinDegrees[x], -1 ≤ x ≤ 1}, x]ArcSinDegrees is injective:
FunctionInjective[ArcSinDegrees[x], x]Plot[{ArcSinDegrees[x], 1 / 2}, {x, -2π, 2π}]ArcSinDegrees is not surjective:
FunctionSurjective[ArcSinDegrees[x], x]Plot[{ArcSinDegrees[x], 2}, {x, -3, 3}]ArcSinDegrees is neither non-negative nor non-positive:
FunctionSign[ArcSinDegrees[x], x]ArcSinDegrees has both singularity and discontinuity in (-∞,-1] and [1,∞):
FunctionSingularities[ArcSinDegrees[x], x]FunctionDiscontinuities[ArcSinDegrees[x], x]ArcSinDegrees is neither convex nor concave:
FunctionConvexity[ArcSinDegrees[x], x]ArcSinDegrees is concave for x in [-1,0]:
FunctionConvexity[{ArcSinDegrees[x], -1 <= x <= 0}, x]Plot[ArcSinDegrees[x], {x, -1, 0}]TraditionalForm formatting:
ArcSinDegrees[α]//TraditionalFormDifferentiation (3)
derivative:Integration (2)
Indefinite integral of ArcSinDegrees:
Integrate[ArcSinDegrees[x], x]Definite integral of ArcSinDegrees over an interval centered at the origin is 0:
Integrate[ArcSinDegrees[x], {x, -1, 1}]Series Expansions (5)
Find the Taylor expansion using Series:
Series[ArcSinDegrees[x], {x, 0, 7}]Plot the first three approximations for ArcSinDegrees around 
:
terms = Normal@Table[Series[ArcSinDegrees[x], {x, 0, m}], {m, 1, 5, 2}];
Plot[{ArcSinDegrees[x], terms}, {x, -1, 1}, PlotRange -> {{-1, 1}, All}]Asymptotic expansion at Infinity:
Series[ArcSinDegrees[x], {x, ∞, 5}]Asymptotic expansion at a singular point:
Series[ArcSinDegrees[x], {x, -1, 5}]Find series expansions at branch points and branch cuts:
Series[ArcSinDegrees[x], {x, 1, 1}]Series[ArcSinDegrees[x], {x, -2, 1}]ArcSinDegrees can be applied to power series:
ArcSinDegrees[x + (x^2/2) + (x^3/3) + O[x]^4]Function Identities and Simplifications (2)
Simplify expressions involving ArcSinDegrees:
FullSimplify[ArcSinDegrees[Sqrt[1 - x^2]], 0 ≤ x ≤ 1]Use TrigToExp to express through logarithms and square roots:
TrigToExp@ArcSinDegrees[z]Function Representations (1)
Represent using ArcCscDegrees:
ArcCscDegrees[1 / x]//FullSimplifyApplications (9)
Solve an inverse trigonometric equation:
Solve[ArcSinDegrees[z]^2 + 3 ArcSinDegrees[z] == 2, z]Solve an inverse trigonometric equation with a parameter:
Reduce[ArcSinDegrees[SinDegrees[z]] == w, z]Get the zeros of ArcSinDegrees:
Reduce[ArcSinDegrees[α x + β] == 0, x]Use Reduce to solve inequalities involving ArcSinDegrees:
Reduce[ArcSinDegrees[x] > 60, x]Numerically find a root of a transcendental equation:
FindRoot[ArcSinDegrees[z] + 15 ArcSinDegrees[z + 1 / 3] == 3, {z, -1, 1}]//ChopPlot the function to check if the solution is correct:
Plot[ArcSinDegrees[z] + 15 ArcSinDegrees[z + 1 / 3] - 3, {z, -1, 1}]Plot the real and imaginary parts of ArcSinDegrees:
ReImPlot[ArcSinDegrees[x], {x, -2, 2}]Plot the Riemann surface of ArcSinDegrees:
ParametricPlot3D[Evaluate[{Re[SinDegrees[u + I v]], Im[SinDegrees[u + I v]], u}], {u, 0, 720}, {v, -450, 450}]Find the angle between two 3D vectors:
With[{v = Normalize[{1, 1, -2}], u = Normalize[{1, 1, 1}]}, ArcSinDegrees[Norm[Cross[v, u]]]]Different combinations of ArcSinDegrees with trigonometric functions:
{TanDegrees[ArcSinDegrees[z]], TanDegrees[ArcSinDegrees[1 / z]], CotDegrees[ArcSinDegrees[z]], SinDegrees[ArcSinDegrees[z ^ 2]]}Properties & Relations (5)
Compositions with the inverse trigonometric functions:
{SinDegrees[ArcSinDegrees[z]], ArcSinDegrees[SinDegrees[z]]}Use PowerExpand to disregard multivaluedness of the ArcSinDegrees:
PowerExpand[%]Alternatively, evaluate under additional assumptions:
Refine[ArcSinDegrees[SinDegrees[z]], 0 < z < 90]This shows the branch cuts of the ArcSinDegrees function:
Plot3D[Im[ArcSinDegrees[x + I y]], {x, -2, 2}, {y, -2, 2}]ArcSinDegrees gives the angle in degrees, while ArcSin gives the same angle in radians:
ArcSinDegrees[1]ArcSin[1]FunctionExpand applied to ArcSinDegrees generates expressions in trigonometric functions in radians:
FunctionExpand[ArcSinDegrees[x]]FunctionExpand[ArcSinDegrees[x ^ 2]ArcSinDegrees[120 - x / 2]]ExpToTrig applied to the outputs of TrigToExp will generate trigonometric functions in radians:
ArcSinDegrees[z]//TrigToExpExpToTrig[%]Possible Issues (3)
ArcSinDegrees[SinDegrees[30.]]SinDegrees[ArcSinDegrees[30.]]On branch cuts, machine-precision inputs can give numerically wrong answers:
ArcSinDegrees[-2 + I ((Pi + 1) ^ 2 - Pi ^ 2 - 2Pi - 1 - Exp[-Pi ^ 4])]//NArcSinDegrees[-2 + I ((Pi + 1) ^ 2 - Pi ^ 2 - 2Pi - 1 - Exp[-Pi ^ 4])]//N[#, 30]&The precision of the output can be much lower than the precision of the input:
ArcSinDegrees[0.99999999999999999999999999999999999999999999]Neat Examples (3)
Solve trigonometric equations involving ArcSinDegrees:
Reduce[ArcSinDegrees[z] + ArcSinDegrees[z - 1 / 2] == 90, z]//QuietNumerical value of this angle in degrees:
%//NCalculate numerical values of ArcSinDegrees by iteration:
ArcSindeg[x_ ? (InexactNumberQ[#] && 0 < # < 1&)] := FixedPoint[Function[ξ, ξ + 1 / CosDegrees[ξ](x - SinDegrees[ξ])], x]{ArcSindeg[N[7 / 11, 20]], ArcSinDegrees[N[7 / 11, 20]]}Plot ArcSinDegrees at integer points:
ArrayPlot[Table[FractionalPart[Abs[ArcSinDegrees[x y]]], {x, -40, 40}, {y, -40, 40}]]Text
Wolfram Research (2024), ArcSinDegrees, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcSinDegrees.html.
CMS
Wolfram Language. 2024. "ArcSinDegrees." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ArcSinDegrees.html.
APA
Wolfram Language. (2024). ArcSinDegrees. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcSinDegrees.html