ArcCscDegrees
gives the arc cosecant in degrees of the complex number 
.
Details
- ArcCscDegrees, 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 z outside the interval
to 
, the results are always in the range 
to 
, excluding 0. - ArcCscDegrees[z] returns the angle
in degrees for which the ratio of the hypotenuse to the opposite side of a right triangle is 
. - For certain special arguments, ArcCscDegrees automatically evaluates to exact values.
- ArcCscDegrees can be evaluated to arbitrary numerical precision.
- ArcCscDegrees automatically threads over lists.
- ArcCscDegrees[z] has a branch cut discontinuity in the complex
plane running from 
to 
. - ArcCscDegrees 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)
ArcCscDegrees[1]Calculate the angle ABC of this right triangle:
β = ArcCscDegrees[8 / 3]The numerical value of this angle:
%//NSolve an inverse trigonometric equation:
Solve[ArcCscDegrees[x] == 60, x]Solve an inverse trigonometric inequality:
Reduce[ArcCscDegrees[x] > 60, x]Apply ArcCscDegrees to the following list:
ArcCscDegrees[{1, Sqrt[2] (-1 + Sqrt[3]), (2/Sqrt[3]), Sqrt[2], 2, Sqrt[2] (1 + Sqrt[3])}]Plot over a subset of the reals:
Plot[ArcCscDegrees[x], {x, -5, 5}]Asymptotic expansion at Infinity:
Series[ArcCscDegrees[x], {x, ∞, 5}]Scope (38)
Numerical Evaluation (6)
ArcCscDegrees[5.0]N[ArcCscDegrees[5], 50]The precision of the output tracks the precision of the input:
ArcCscDegrees[5.00000000000000000000000000000000000000000]Evaluate for complex arguments:
ArcCscDegrees[0.5 + 2.3I]Evaluate ArcCscDegrees efficiently at high precision:
ArcCscDegrees[5`500]//TimingArcCscDegrees[5`100000];//TimingCompute worst-case guaranteed intervals using Interval and CenteredInterval objects:
ArcCscDegrees[Interval[{1, 5}]]ArcCscDegrees[CenteredInterval[-3, 1 / 100]]ArcCscDegrees[CenteredInterval[2 + 3I, (1 + I) / 100]]Or compute average-case statistical intervals using Around:
ArcCscDegrees[Around[(2/Sqrt[3]), 0.01]]Compute the elementwise values of an array:
ArcCscDegrees[{{Sqrt[2], -Sqrt[2]}, {0, -Sqrt[2]}}]Or compute the matrix ArcCscDegrees function using MatrixFunction:
MatrixFunction[ArcCscDegrees[#]&, {{Sqrt[2], -Sqrt[2]}, {0, -Sqrt[2]}}]Specific Values (5)
Values of ArcCscDegrees at fixed points:
ArcCscDegrees[{-Sqrt[2], -(2/Sqrt[3]), (2/Sqrt[3]), Sqrt[2]}]Simple exact values are generated automatically:
ArcCscDegrees[(2/Sqrt[3])]ArcCscDegrees[Infinity]ArcCscDegrees[ComplexInfinity]Singular points of ArcCscDegrees:
ArcCscDegrees[{1, -1, 0}]ArcCscDegrees is not differentiable at these points:
ArcCscDegrees'[{1, -1, 0}]//QuietFind the value of 
satisfying equation 
:
f[x_] := ArcCscDegrees[x] - 60;sol = Solve[f[x] == 0, x]xzero = x /. First[sol]Plot[f[x], {x, 1, 2}, Rule[...]]Visualization (4)
Plot the ArcCscDegrees function:
Plot[ArcCscDegrees[x], {x, -2, 2}]Plot over a subset of the complexes:
ComplexPlot3D[ArcCscDegrees[z], {z, -4 - 2I, 4 + 2I}, Rule[...]]Plot the real part of ArcCscDegrees:
ComplexContourPlot[Re[ArcCscDegrees[z]], {z, -2π - 2π I, 2π + 2π I}, IconizedObject[«PlotOptions»]]Plot the imaginary part of ArcCscDegrees:
ComplexContourPlot[Im[ArcCscDegrees[z]], {z, -2π - 2π I, 2π + 2π I}, IconizedObject[«PlotOptions»]]Polar plot with ArcCscDegrees:
Table[PolarPlot[ArcCscDegrees[k ϕ], {ϕ, -π, π}, ...], {k, 1, 4}]Function Properties (11)
ArcCscDegrees is defined for all real values except from the interval 
:
FunctionDomain[ArcCscDegrees[x], x]FunctionDomain[ArcCscDegrees[z], z, Complexes]ArcCscDegrees achieves all real values from the interval 
except 
:
FunctionRange[ArcCscDegrees[x], x, y]FunctionRange[ArcCscDegrees[x], x, y, Complexes]ArcCscDegrees is an odd function:
ArcCscDegrees[-x]ArcCscDegrees is not an analytic function:
FunctionAnalytic[ArcCscDegrees[x], x]FunctionMeromorphic[ArcCscDegrees[x], x]ArcCscDegrees is monotonic in a specific range:
FunctionMonotonicity[ArcCscDegrees[x], x]FunctionMonotonicity[{ArcCscDegrees[x], x ≤ -1 }, x]FunctionMonotonicity[{ArcCscDegrees[x], x ≥ 1}, x]Plot[ArcCscDegrees[x], {x, -3, 3}]ArcCscDegrees is injective:
FunctionInjective[ArcCscDegrees[x], x]Plot[{ArcCscDegrees[x], 1 / 2}, {x, -5, 5}]ArcCscDegrees is not surjective:
FunctionSurjective[ArcCscDegrees[x], x]Plot[{ArcCscDegrees[x], 120}, {x, -5, 5}]ArcCscDegrees is neither non-negative nor non-positive:
FunctionSign[ArcCscDegrees[x], x]It has both singularity and discontinuity for x in [-1,1]:
FunctionSingularities[ArcCscDegrees[x], x]FunctionDiscontinuities[ArcCscDegrees[x], x]ArcCscDegrees is neither convex nor concave:
FunctionConvexity[ArcCscDegrees[x], x]ArcCscDegrees is convex for x in [1,∞):
FunctionConvexity[{ArcCscDegrees[x], x >= 1}, x]Plot[ArcCscDegrees[x], {x, 1, 10}]TraditionalForm formatting:
ArcCscDegrees[α]//TraditionalFormDifferentiation (3)
D[ArcCscDegrees[x], x]Table[D[ArcCscDegrees[x], {x, n}], {n, 1, 4}]//FullSimplifyPlot[Evaluate[%], {x, -3, 3}, Rule[...]]
derivative:D[ArcCscDegrees[x], {x, n}]Integration (2)
Indefinite integral of ArcCscDegrees:
Integrate[ArcCscDegrees[x], x]Definite integral of ArcCscDegrees over the interval 
:
Integrate[ArcCscDegrees[x], {x, 1, 2}]Series Expansions (4)
Find the Taylor expansion using Series:
Series[ArcCscDegrees[x], {x, 2, 4}]Plots of the first three approximations for ArcCscDegrees around 
:
terms = Normal@Table[Series[ArcCscDegrees[x], {x, 2, m}], {m, 1, 3}];
Plot[{ArcCscDegrees[x], terms}, {x, 1, 3}, PlotRange -> All]Find series expansions at branch points and branch cuts:
Series[ArcCscDegrees[x], {x, 1, 1}]Series[ArcCscDegrees[x], {x, -1 / 2, 1}]Asymptotic expansion at a singular point:
Series[ArcCscDegrees[x], {x, 1, 3}, Assumptions -> x > 1]//FullSimplifyArcCscDegrees can be applied to power series:
ArcCscDegrees[x + x ^ 2 + O[x, Infinity] ^ 4]Function Identities and Simplifications (2)
Simplify expressions involving ArcCscDegrees:
FullSimplify[ArcCscDegrees[Sqrt[(1/1 - x^2)]], 0 ≤ x ≤ 1]Use TrigToExp to express through logarithms and square roots:
ArcCscDegrees[z]//TrigToExpFunction Representations (1)
Represent using ArcSinDegrees:
ArcSinDegrees[(1/z)]//FullSimplifyApplications (6)
Solve inverse trigonometric equations:
Solve[ArcCscDegrees[α x + β] == 4, x]Solve[ArcCscDegrees[z]^2 + 3 ArcCscDegrees[z] == 2, z]Solve an inverse trigonometric equation with a parameter:
Reduce[ArcCscDegrees[CscDegrees[z]] == w, z]Use Reduce to solve inequalities involving ArcCscDegrees:
Reduce[ArcCscDegrees[x] > 60, x]Numerically find a root of a transcendental equation:
FindRoot[ArcCscDegrees[z] + 2 ArcCscDegrees[z + 1 / 3] == 3, {z, 10, 100}]//ChopPlot the function to check if the solution is correct:
Plot[ArcCscDegrees[z] + 2 ArcCscDegrees[z + 1 / 3] - 3, {z, 10, 100}]Plot the real and imaginary parts of ArcCscDegrees:
ReImPlot[ArcCscDegrees[x], {x, -3, 3}]Different combinations of ArcCscDegrees with trigonometric functions:
{TanDegrees[ArcCscDegrees[z]], TanDegrees[ArcCscDegrees[1 / z]], CotDegrees[ArcCscDegrees[z]], SinDegrees[ArcCscDegrees[z]]}Properties & Relations (6)
Compositions with the inverse trigonometric functions:
{ArcCscDegrees[CscDegrees[z]], CscDegrees[ArcCscDegrees[z]]}Use PowerExpand to disregard multivaluedness of the ArcCscDegrees:
PowerExpand[%]Alternatively, evaluate under additional assumptions:
Refine[ArcCscDegrees[CscDegrees[z]], 0 < z < 90]Use FunctionExpand to convert trigs of arctrigs into an algebraic function:
FunctionExpand[TanDegrees[ArcCscDegrees[z]]]FullSimplify[%]This shows the branch cut of the ArcCscDegrees function:
Plot3D[Im[ArcCscDegrees[x - I y]], {x, -2, 2}, {y, -2, 2}]ArcCscDegrees gives the angle in degrees, while ArcCsc gives the same angle in radians:
ArcCscDegrees[2]ArcCsc[2]FunctionExpand applied to ArcCscDegrees generates expressions in trigonometric functions in radians:
FunctionExpand[ArcCscDegrees[x]]FunctionExpand[ArcCscDegrees[x ^ 2]ArcCscDegrees[120 - x / 2]]ExpToTrig applied to the outputs of TrigToExp will generate trigonometric functions in radians:
ArcCscDegrees[z]//TrigToExpExpToTrig[%]Neat Examples (2)
Solve trigonometric equations involving ArcCscDegrees:
Reduce[ArcCscDegrees[z] + ArcCscDegrees[z + 1] == 30, z]//QuietNumerical value of this angle in degrees:
%//NPlot ArcCscDegrees at integer points:
ArrayPlot[Table[FractionalPart[Abs[ArcCscDegrees[x y]]], {x, -40, 40}, {y, -40, 40}]]Text
Wolfram Research (2024), ArcCscDegrees, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCscDegrees.html.
CMS
Wolfram Language. 2024. "ArcCscDegrees." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ArcCscDegrees.html.
APA
Wolfram Language. (2024). ArcCscDegrees. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcCscDegrees.html