Csc

Cosecant, the reciprocal of Sin.

✓ checked against that system · ! checked, with disagreements (hover a chip)DLMF4.14.E5Wolfram LanguageCsc!

Csc(x)the cosecant of x,

\frac{1}{\sin(x)}

.

SageMathcsc(x)SymPycsc(x)mpmathcsc(x)Juliaenumeratio_ball(_csc, x)

Typed signatures

Type	Package	Applies
`(complex \| quantity) -> number`	`analytic`replaces `compute-engine`	always
`(complex \| quantity) -> number`	`compute-engine`	always

Domain: Elementary functions

Details

$\csc(x) = \frac{1}{\sin(x)}$ .
Odd function, period $2\pi$ , same as Sin.
Undefined wherever $\sin(x) = 0$ , the same poles as Cot.

Examples

6 more kept as data, checked against the oracles

Basic #

In

\csc(\frac{π}{6})

Out

2

In

\csc(\frac{π}{4})

Out

\sqrt{2}

In

\csc(\frac{π}{2})

Out

1

$\csc 60^\circ = \frac{2\sqrt3}{3}$ , exactly

In

\csc(60^{∘})

Out

\frac{2\sqrt{3}}{3}

A machine-precision argument evaluates numerically

In

\csc(1.2)

Out

1.072\,916\,377\,709\,897\,228\,71

In

\operatorname{Series}(\csc(x), x, 0, 10)

Out

\frac{1}{x}+\frac{x}{6}+\frac{7x^3}{360}+\frac{31x^5}{15\,120}+\frac{127x^7}{604\,800}+\frac{73x^9}{3\,421\,440}+O\left(x^{11}\right)

In

\operatorname{Series}(\csc(x), x, π, 5)

Out

-\frac{1}{x-π}-\frac{x-π}{6}-\frac{1}{360}×7(x-π)^3-\frac{1}{15\,120}×31(x-π)^5+O\left((x-π)^7\right)

Scope #

N(...) of an exact argument gives an arbitrary-precision value

In

N(\csc(\frac{6}{5}))

Out

1.072\,916\,377\,709\,897\,228\,71

Evaluates at a complex argument

In

\csc(2.5+i)

Out

0.530\,965\,577\,621\,174\,2+0.541\,322\,906\,190\,002\,5i

Threads element-wise over a matrix, with ComplexInfinity at the pole $\pi$

In

\csc(\bigl\lbrack\bigl\lbrack\frac{π}{3}, π\bigr\rbrack, \bigl\lbrack\frac{π}{6}, -π/2\bigr\rbrack\bigr\rbrack)

Out

\bigl\lbrack\bigl\lbrack\frac{2\sqrt{3}}{3}, \tilde∞\bigr\rbrack, \bigl\lbrack2, -1\bigr\rbrack\bigr\rbrack

$\csc(\pi/12) = \sqrt6 + \sqrt2$

In

\csc(\frac{π}{12})

Out

\sqrt{2}+\sqrt{6}

$\csc(\pi/10) = 1 + \sqrt5$

In

\csc(\frac{π}{10})

Out

1+\sqrt{5}

In

\csc(\frac{π}{5})

Out

\frac{1}{5}\sqrt{10}\sqrt{5+\sqrt{5}}

Interval arithmetic: the image of $[\pi/6, \pi/3]$ is $[\tfrac{2}{\sqrt3}, 2]$

In

\csc(\mathrm{Interval}(\frac{π}{6}, \frac{π}{3}))

Out

\mathrm{Interval}(\frac{2\sqrt{3}}{3}, 2)

Uncertainty propagation: $\csc(2 \pm 0.01) \approx 1.0998 \pm 0.0050$

In

\csc(\mathrm{Around}(2, 0.01))

Out

\mathrm{Around}(1.099\,750\,170\,294\,616\,4, 0.005\,033\,089\,733\,443\,42)

Oddness pulls the sign out of a symbolic argument: $\csc(-x) = -\csc x$

In

\csc(-x)

Out

-\csc(x)

Derivative: $-\csc x \cot x$

In

\frac{\mathrm{d}\csc(x)}{\mathrm{d}x}

Out

-(\csc(x)\cot(x))

Antiderivative: $-\ln|\csc x + \cot x|$

In

\int\!\csc(x)\, \mathrm{d}x

Out

-\ln(\vert\csc(x)+\cot(x)\vert)

Fifty digits of a rational argument

In

N(\csc(\frac{6}{5}), 50)

Out

1.072\,916\,377\,709\,897\,228\,705\,116\,922\,435\,239\,154\,837\,639\,006\,697\,8

Matrix function with exact rational and radical entries

In

\mathrm{MatrixFunction}(\operatorname{\_1}↦\csc(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack\frac{π}{3}, π\bigr\rbrack, \bigl\lbrack\frac{π}{6}, -\frac{π}{2}\bigr\rbrack\bigr\rbrack)

Out

\bigl\lbrack\bigl\lbrack\frac{6}{7}-\frac{2\sqrt{3}}{21}, \frac{6}{7}+\frac{4\sqrt{3}}{7}\bigr\rbrack, \bigl\lbrack\frac{1}{7}+\frac{2\sqrt{3}}{21}, \frac{1}{7}-\frac{4\sqrt{3}}{7}\bigr\rbrack\bigr\rbrack

$\csc(\infty)$ is a type error at boxing; Wolfram gives the oscillation range, $(-\infty, -1] \cup [1, \infty)$

In

\mathrm{Assuming}(m∈ℤ, \mathrm{Refine}(\csc(πm)))

Out

\csc(πm)

In

\mathrm{Assuming}(m∈ℤ, \mathrm{FullSimplify}(\mathrm{Refine}(\csc(π(1/2+m)))))

Out

\csc(π(m+\frac{1}{2}))

In

\mathrm{FunctionMeromorphic}(\csc(x), x)

Out

\mathrm{True}

Properties #

Reciprocal of Sin: $\csc(x) = \frac{1}{\sin(x)}$

In

\csc(\frac{π}{6})=\frac{1}{\sin(\frac{π}{6})}

Out

\mathrm{True}

$\csc(\pi/8)$ as a nested radical

In

\csc(\frac{π}{8})

Out

\sqrt{4+2\sqrt{2}}

Known value · Wolfram MathematicalFunctionData, Csc

$\csc(3\pi/10) = \sqrt5 - 1$

In

\csc(\frac{3π}{10})

Out

\sqrt{5}-1

Known value · Wolfram MathematicalFunctionData, Csc

In

\csc(\frac{1}{2})^8-2(2\csc(1/2)^2-2)\csc(1)^2\csc(\frac{1}{2})^4

Out

\csc(\frac{1}{2})^8-4(\csc(1/2)^2-1)\csc(1)^2\csc(\frac{1}{2})^4

Known value · Wolfram MathematicalFunctionData, Csc, FunctionalEquations

In

N(4(\csc(1/2)^2-1)\csc(1)^2)

Out

18.928\,462\,573\,893\,558\,010\,575\,088\,586\,931\,068\,594\,797\,6

Known value · Wolfram MathematicalFunctionData, Csc, FunctionalEquations

In

N(\csc(1/2+π/2)^2)

Out

1.298\,446\,410\,409\,524\,836\,88

Known value · Wolfram MathematicalFunctionData, Csc, FunctionalEquations

In

N(4(\csc(2)^2-1)\csc(4)^2)

Out

1.462\,770\,359\,711\,973\,446\,186\,309\,480\,011\,456\,569\,948

Known value · Wolfram MathematicalFunctionData, Csc, FunctionalEquations

Possible issues #

A pole: $\sin(0) = 0$

In

\csc(0)

Out

\tilde∞

Csc(ComplexInfinity) is Indeterminate, for the same recurring poles as Tan

Known value · DLMF 4.14.10

About

Examples	39
Checked against	Wolfram Language 30 agree8 differ1 unevaluatedSageMath 29 agree1 unevaluated2 errorSymPy 31 agree3 differmpmath 22 agree3 differ1 errorJulia 23 agree3 errorRust 16 agree7 differ1 error
Crosswalk	DLMF Wolfram LanguageSageMathSymPympmathJulia

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

The reciprocal of sin.

See also: Sin, Sec, Cot