Csc
Csc[z]
gives the cosecant of z.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The argument of Csc is assumed to be in radians. (Multiply by Degree to convert from degrees.)
. - 1/Sin[z] is automatically converted to Csc[z]. TrigFactorList[expr] does decomposition.
- For certain special arguments, Csc automatically evaluates to exact values.
- Csc can be evaluated to arbitrary numerical precision.
- Csc automatically threads over lists. »
- Csc can be used with Interval and CenteredInterval objects. »
Background & Context
- Csc is the cosecant function, which is one of the basic functions encountered in trigonometry. It is defined as the reciprocal of the sine function:
. It is defined for real numbers by letting 
be a radian angle measured counterclockwise from the 
axis along the circumference of the unit circle. Csc[x] then gives the reciprocal of the vertical coordinate of the arc endpoint. The equivalent schoolbook definition of the cosecant of an angle 
in a right triangle is the ratio of the length of the hypotenuse to the length of the leg opposite 
. - Csc automatically evaluates to exact values when its argument is a simple rational multiple of
. For more complicated rational multiples, FunctionExpand can sometimes be used to obtain an explicit exact value. TrigFactorList can be used to factor expressions involving Csc into terms containing Sin and Cos. To specify an argument using an angle measured in degrees, the symbol Degree can be used as a multiplier (e.g. Csc[30 Degree]). When given exact numeric expressions as arguments, Csc may be evaluated to arbitrary numeric precision. Other operations useful for manipulation of symbolic expressions involving Csc include TrigToExp, TrigExpand, Simplify, and FullSimplify. - Csc threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the cosecant of a square matrix (i.e. the power series for the cosecant function with ordinary powers replaced by matrix powers) as opposed to the cosecants of the individual matrix elements.
- Csc is periodic with period
, as reported by FunctionPeriod. Csc satisfies the identity 
, which is equivalent to the Pythagorean theorem. The definition of the cosecant function is extended to complex arguments 
using the definition 
, where 
is the base of the natural logarithm. Csc has poles at 
for 
an integer and evaluates to ComplexInfinity at these points. Csc[z] has series expansion ![sum_(k=0)^infty((-1)^(k+1) 2(2^(2 k-1)-1) TemplateBox[{{2, , k}}, BernoulliB] )/((2 k)!)z^(2 k-1)](Files/Csc.en/15.png "sum_(k=0)^infty((-1)^(k+1) 2(2^(2 k-1)-1) TemplateBox[{{2, , k}}, BernoulliB] )/((2 k)!)z^(2 k-1)")
about the origin that may be expressed in terms of the Bernoulli numbers BernoulliB. - The inverse function of Csc is ArcCsc. The hyperbolic cosecant is given by Csch. Other related mathematical functions include Sec and Sin.
Examples
open all close allBasic Examples (6)
The argument is given in radians:
Csc[Pi / 3]Use Degree to specify an argument in degrees:
Csc[60Degree]Plot over a subset of the reals:
Plot[Csc[x], {x, -Pi / 2, 3 / 2Pi}]Plot over a subset of the complexes:
ComplexPlot3D[Csc[z], {z, -π - I, π + I}, PlotLegends -> Automatic]Series[Csc[x], {x, 0, 10}]Asymptotic expansion at a singular point:
Series[Csc[x], {x, π, 5}]Scope (47)
Numerical Evaluation (6)
Csc[1.2]N[Csc[12 / 10], 50]The precision of the output tracks the precision of the input:
Csc[1.2000000000000000000000000000000000000000000]Evaluate for complex arguments:
Csc[2.5 + I]Evaluate Csc efficiently at high precision:
Csc[1.2`500]//TimingCsc[1.2`100000];//TimingCompute the elementwise values of an array using automatic threading:
Csc[ {{π / 3, π}, {π / 6, -π / 2}}]Or compute the matrix Csc function using MatrixFunction:
MatrixFunction[Csc[#]&, {{π / 3, π}, {π / 6, -π / 2}}]Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:
Csc[Interval[{Pi / 6, Pi / 3}]]Csc[CenteredInterval[1, 1 / 100]]Csc[CenteredInterval[2 + 3I, (1 + I) / 100]]Or compute average-case statistical intervals using Around:
Csc[Around[2, 0.01]]Specific Values (6)
Values of Csc at fixed points:
Table[Csc[n (π/6)], {n, 1, 12, 2}]Csc[Infinity]Csc[ComplexInfinity]Singular points of Csc:
Assuming[m∈Integers, Refine[Csc[π m]]]Local extrema of Csc:
Assuming[m∈Integers, FullSimplify[Refine[Csc[π ((1/2) + m)]]]]Find a local minimum of Csc as the root of 
in the minimum's neighborhood:
sol = Solve[D[Csc[x], x] == 0 && 0 < x < π, x]xmin = x /. First[sol]Plot[Csc[x], {x, 0, π}, Epilog -> Style[Point[{xmin, Csc[xmin]}], PointSize[Large], Red]]Simple exact values are generated automatically:
Csc[Pi / 5]More complicated cases require explicit use of FunctionExpand:
Csc[Pi / 30]FunctionExpand[%]Visualization (3)
Plot the Csc function:
Plot[Csc[x], {x, 0, 4π}]
ComplexContourPlot[Re[Csc[z]], {z, -π - π I, π + π I}, IconizedObject[«PlotOptions»]]
ComplexContourPlot[Im[Sec[z]], {z, -π - π I, π + π I}, IconizedObject[«PlotOptions»]]
Table[PolarPlot[Csc[k ϕ], {ϕ, -2π, 2π}, Sequence[Frame -> True, Exclusions -> All, PlotLabel -> "k=" <> ToString[k], PlotRange -> {{-5, 5}, {-5, 5}}]], {k, 1, 6}]Function Properties (13)
The real domain of Csc:
FunctionDomain[Csc[x], x]FunctionDomain[Csc[z], z, Complexes]Csc achieves all real values except from the open interval 
:
FunctionRange[Csc[x], x, y]Csc is a periodic function with a period 
:
FunctionPeriod[Csc[x], x]Csc is an odd function:
Csc[-x]Csc has the mirror property ![csc(TemplateBox[{z}, Conjugate])=TemplateBox[{{csc, (, z, )}}, Conjugate]](Files/Csc.en/22.png "csc(TemplateBox[{z}, Conjugate])=TemplateBox[{{csc, (, z, )}}, Conjugate]"){kind=small}
:
FullSimplify[Csc[Conjugate[z]] == Conjugate[Csc[z]]]Csc is not an analytic function:
FunctionAnalytic[Csc[x], x]FunctionMeromorphic[Csc[x], x]Csc is monotonic in a specific range:
FunctionMonotonicity[Csc[x], x]FunctionMonotonicity[{Csc[x], 0 < x < π / 2}, x]Csc is not injective:
FunctionInjective[Csc[x], x]Plot[{Csc[x], 2}, {x, -2π, 2π}]Csc is not surjective:
FunctionSurjective[Csc[x], x]Plot[{Csc[x], .5}, {x, -2π, 2π}]Csc is neither non-negative nor non-positive:
FunctionSign[Csc[x], x]It has both singularity and discontinuity when x is a multiple of π:
FunctionSingularities[Csc[x], x]FunctionDiscontinuities[Csc[x], x]FunctionConvexity[Csc[x], x]FunctionConvexity[{Csc[x], 0 < x < 3}, x]Plot[Csc[x], {x, 0, 3}]TraditionalForm formatting:
Csc[z]//TraditionalFormDifferentiation (3)
D[Csc[x], x]Table[D[Csc[x], {x, n}], {n, 1, 4}]Plot[Evaluate[%], {x, -2π, 2π}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative", "Fourth Derivative"}]
derivative:D[Csc[x], {x, n}]Integration (3)
Indefinite integral of Csc:
Integrate[Csc[x], x]Definite integral of Csc over a period is 0:
Integrate[Csc[x], {x, π / 6, π / 6 + 2π}, PrincipalValue -> True]Integrate[Csc[x]Cos[x], x]Integrate[Csc[z]^a, z]Series Expansions (3)
Find the Taylor expansion using Series:
Series[Csc[x], {x, π / 2, 8}]Plot the first three approximations for Csc around 
:
terms = Normal@Table[Series[Csc[x], {x, π / 2, m}], {m, 2, 6, 2}];
Plot[{Csc[x], terms}, {x, 0, π}, PlotRange -> {0, 4}]General term in the series expansion of Csc:
SeriesCoefficient[Csc[x], {x, 0, n}]Csc can be applied to power series:
Csc[(π/2) + x + (x^2/2) + (x^3/3) + O[x]^4]Function Identities and Simplifications (6)
Csc of a double angle:
Expand[Csc[2x] , Trig -> True]Csc of a sum:
Expand[Csc[x + y] , Trig -> True]Convert multiple‐angle expressions:
TrigExpand[Csc[4x]]TrigReduce[%]Convert sums of trigonometric functions to products:
TrigFactor[Csc[x] + Csc[y]]Expand assuming real variables 
and 
:
ComplexExpand[Csc[x + I y]]Convert to complex exponentials:
TrigToExp[Csc[z]]Function Representations (4)
Representation through Sin:
Simplify[1 / Sin[x]]Representation through Bessel functions:
Simplify[Sqrt[ 2 / π / x] / BesselJ[(1/2), x], x ≥ 0]Simplify[I Sqrt[ 2 / (π I x)] / BesselI[(1/2), I x], x ≥ 0]Representation through SphericalHarmonicY:
Simplify[Sqrt[3 / 8 / π] / SphericalHarmonicY[1, -1, θ, 0]]Representation in terms of MeijerG:
1 / MeijerGReduce[Sin[x], x]Activate[%]Applications (3)
Generate a plot with poles automatically removed:
Plot[Csc[x], {x, -2, 6}]Generate a plot over the complex argument plane:
Plot3D[Re[Csc[x + I y]], {x, -Pi, 2Pi}, {y, 0, 3}]Solve a differential equation:
DSolve[(a + x Cot[x]) y[x] + (x + 2 x^2 Cot[x]) Derivative[1][y][x] + x^2 Derivative[2][y][x] == 0, y[x], x]Properties & Relations (12)
Basic parity and periodicity properties of the cosecant function get automatically applied:
Csc[x + Pi]Csc[-x]Csc[I x]1 / Csc[x]Use TrigFactorList to factor Csc into Sin and Cos:
1 / Sin[x]TrigFactorList[Csc[x]]Complicated expressions containing trigonometric functions do not automatically simplify:
1 / Csc[Subscript[z, 1] + Subscript[z, 2]] - 2Sin[Subscript[z, 2]]Cos[Subscript[z, 1]]Simplify[%]Csc[Conjugate[z]] - Conjugate[Csc[z]]FullSimplify[%]Simplification with additional assumptions:
Csc[-x + k Pi]Simplify[%, k∈Integers]Compositions with the inverse functions:
{Csc[ArcCsc[z]], Csc[2ArcCsc[z]], Csc[3ArcCsc[z]]}FunctionExpand[%]//Together
Csc[1] == CscDegrees[180 / π]//SimplifySolve a trigonometric equation:
Reduce[3Csc[z]^2 - 6Csc[z - Pi / 6] == -8, z]Reduce[Csc[α x + β] == 0, x]Reduce[1 / Csc[α x + β] == 0, x]Numerically find a root of a transcendental equation:
FindRoot[Csc[z]^2 + 3 Csc[z + Pi / 3] + z == 4, {z, 5}]Csc is automatically returned as a special case for many mathematical functions:
{BesselJ[1 / 2, z]^-1, StruveH[-1 / 2, z]^-1, 1 / MathieuS[1, 0, z], JacobiDS[z, 0], JacobiNS[z, 0], HypergeometricPFQ[{}, {(3/2)}, -z]^-1, MeijerG[{{}, {}}, {{(1/2)}, {0}}, z]^-1}Calculate residue symbolically and numerically:
Table[Residue[Csc[z]^k, {z, 0}], {k, 5}](1/2π I)NIntegrate[Csc[z], {z, -(1/4), -(I/4), +(1/4), +(I/4), -(1/4)}]Csc is a numeric function:
Attributes[Csc]NumericQ[Csc[2 + E]]Possible Issues (4)
Machine-precision input is insufficient to give a correct answer:
Csc[10. ^ 30]Use arbitrary-precision evaluation instead:
N[Csc[10 ^ 30], 10]A larger setting for $MaxExtraPrecision is needed to accurately approximate function value:
N[Csc[10 ^ 65], 20]
Block[{$MaxExtraPrecision = 70}, N[Csc[10 ^ 65], 20]]The precision of the output can be much smaller or larger than the precision of the input:
Csc[3.141592653589793238462643383279502884297]Csc[1.5707963267948966192213216916397514421]In traditional form, parentheses are needed around the argument:
csc xcsc(x)Neat Examples (6)
Integrate[Csc[x], x]∫Sin[b z] Csc[c z]ⅆz∫z^2 E^b zCsc[a z]ⅆzCompute a product of cosecants:
Underoverscript[∏, k = 1, n - 1] Csc[z + (k π/n)]Plot Csc at integer points:
ArrayPlot[Table[ArcTan[Abs[Csc[x y]]], {x, -20, 20}, {y, -20, 20}]]Give a nested radical representation:
Csc[(π/2^12)]//FunctionExpandGenerate Csc from integrals and sums:
(1/z)Underoverscript[∏, k = 1, ∞](π^2 k^2/π^2 k^2 - z^2)-2I Underoverscript[∑, k = 1, ∞] (E^I z)^2 k - 1//FullSimplifySum[(I I^-1 + 2 k 2^2 k (-1 + 2^1 - 2 k) z^-1 + 2 k BernoulliB[2 k]/(2 k)!), {k, 1, ∞}]//TrigToExp//FullSimplifyCsc[n] is a transcendental number at integer n:
Element[Csc[1], Algebraics]Related Links
MathWorld
The Wolfram Functions Site
NKS|OnlineText
Wolfram Research (1988), Csc, Wolfram Language function, https://reference.wolfram.com/language/ref/Csc.html (updated 2021).
CMS
Wolfram Language. 1988. "Csc." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Csc.html.
APA
Wolfram Language. (1988). Csc. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Csc.html