Arccsc

Arccosecant, the inverse of Csc.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageArcCsc!↗

Arccsc(x)the value $y$ with $\csc(y) = x$.

SageMatharccsc(x)SymPyacsc(x)mpmathacsc(x)Juliaenumeratio_ball(_acsc, x)Rustasin(n(1) / x)

Typed signatures

Type	Package	Applies
(complex | infinity | quantity) -> number	analyticreplaces compute-engine	always
(complex | infinity | quantity) -> number	compute-engine	always

Domain: Elementary functions

Details

• Differs from Wolfram: past its real domain a rational argument reduces to the exact principal value, on every inverse trigonometric and hyperbolic head alike (Arcsin, Arccos, Arcsec, Arccsc, Arcosh, Artanh, Arcoth, Arsech); Wolfram leaves the exact call unevaluated and gives the same number from N[...]. Checked against mpmath and wolframscript on a grid of rationals to double precision. A float and N(...) already took the same branch. Compiled code is separate: the JavaScript target is real-valued (NaN past the domain), the WGSL target returns a complex pair, and Arcsin and Arccos decline to compile. For $0<|x|<1$, $\operatorname{arccsc}(x) = \arcsin(1/x)$, so $\operatorname{arccsc}(1/2) = \frac{\pi}{2} - i\ln(2+\sqrt3)$ (mpmath acsc(0.5)).

Examples

Basic #

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In$\mathrm{arccsc}(1)$Out$\frac{π}{2}$

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In$\mathrm{arccsc}(2)$Out$\frac{π}{6}$

Scope #

$\operatorname{arccsc}\sqrt2 = \pi/4$

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In$\mathrm{arccsc}(\sqrt{2})$Out$\frac{π}{4}$

An odd function

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In$\mathrm{arccsc}(-2)$Out$-\frac{π}{6}$

Inside $(-1, 1)$ the value is complex

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In$\mathrm{arccsc}(0.5)$Out$1.570\,796\,326\,794\,896\,6-1.316\,957\,896\,924\,816\,6i$

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In$\mathrm{arccsc}(∞)$Out$0$

A pole: $\operatorname{arccsc} 0$ is complex infinity

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In$\mathrm{arccsc}(0)$Out$\tilde∞$

Derivative, for real $|x| > 1$

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In$\frac{\mathrm{d}\mathrm{arccsc}(x)}{\mathrm{d}x}$Out$-\frac{1}{\vert x\vert\sqrt{x^2-1}}$

Fifty digits of a reciprocal-sine angle

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In$N(\mathrm{arccsc}(5), 50)$Out$0.201\,357\,920\,790\,330\,791\,455\,125\,552\,217\,623\,410\,240\,038\,081\,402\,23$

Threads over a matrix; a zero entry gives complex infinity

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In$\mathrm{arccsc}(\bigl\lbrack\bigl\lbrack\sqrt{2}, -2^{1/2}\bigr\rbrack, \bigl\lbrack0, -2^{1/2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{π}{4}, -\frac{π}{4}\bigr\rbrack, \bigl\lbrack\tilde∞, -\frac{π}{4}\bigr\rbrack\bigr\rbrack$

Reciprocal-sine values at $\pm\pi/4$ and $\pm\pi/3$

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In$\bigl\lbrack\mathrm{arccsc}(-\sqrt{2}), \mathrm{arccsc}(-\frac{2\sqrt{3}}{3}), \mathrm{arccsc}(\frac{2\sqrt{3}}{3}), \mathrm{arccsc}(\sqrt{2})\bigr\rbrack$Out$\bigl\lbrack-\frac{π}{4}, -\frac{π}{3}, \frac{π}{3}, \frac{π}{4}\bigr\rbrack$

At complex infinity $1/z \to 0$, so the value is 0

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In$\mathrm{arccsc}(\tilde∞)$Out$0$

Zero has no finite csc preimage and gives complex infinity

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In$\mathrm{arccsc}(\bigl\lbrack1, -1, 0\bigr\rbrack)$Out$\bigl\lbrack\frac{π}{2}, -\frac{π}{2}, \tilde∞\bigr\rbrack$

As a matrix function the zero entry becomes 0, unlike entrywise

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{arccsc}(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack\sqrt{2}, -2^{1/2}\bigr\rbrack, \bigl\lbrack0, -2^{1/2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{π}{4}, -\frac{π}{4}\bigr\rbrack, \bigl\lbrack0, -\frac{π}{4}\bigr\rbrack\bigr\rbrack$

For $0<|x|<1$ the value is $\arcsin(1/x)$: $\operatorname{arccsc}(1/2) = \frac{\pi}{2} - i\ln(2+\sqrt3)$

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In$\mathrm{arccsc}(\frac{1}{2})$Out$\frac{π}{2}-i\ln(2+\sqrt{3})$

Odd: $\operatorname{arccsc}(-1/2)$ is the negative of $\operatorname{arccsc}(1/2)$

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In$\mathrm{arccsc}(-\frac{1}{2})$Out$i\ln(2+\sqrt{3})-\frac{π}{2}$

Properties #

Undoes Csc: $\csc(\text{arccsc}(x)) = x$

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In$\csc(\mathrm{arccsc}(2))=2$Out$\mathrm{True}$

$\csc(-\pi/2) = -1$

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In$\mathrm{arccsc}(-1)$Out$-\frac{π}{2}$

Known value · Wolfram MathematicalFunctionData, ArcCsc

An algebraic argument with value $5\pi/12$

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In$\mathrm{arccsc}(\sqrt{6}-\sqrt{2})$Out$\frac{5π}{12}$

Known value · Wolfram MathematicalFunctionData, ArcCsc

Swapping the terms flips the sign of the angle

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In$\mathrm{arccsc}(\sqrt{2}-\sqrt{6})$Out$-\frac{5π}{12}$

Known value · Wolfram MathematicalFunctionData, ArcCsc

Stays unevaluated; the radical is not matched to a rational multiple of $\pi$

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In$\mathrm{arccsc}(2\sqrt{\frac{1}{2+\sqrt{2}}})$Out$\mathrm{arccsc}(\frac{2}{\sqrt{2+\sqrt{2}}})$

Known value · Wolfram MathematicalFunctionData, ArcCsc

Likewise for the negated argument

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In$\mathrm{arccsc}(-2\sqrt{\frac{1}{2+\sqrt{2}}})$Out$\mathrm{arccsc}(-\frac{2}{\sqrt{2+\sqrt{2}}})$

Known value · Wolfram MathematicalFunctionData, ArcCsc

$\csc(\pi/3) = \frac{2}{\sqrt3}$

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In$\mathrm{arccsc}(\frac{2\sqrt{3}}{3})$Out$\frac{π}{3}$

Known value · Wolfram MathematicalFunctionData, ArcCsc

Odd symmetry gives $-\pi/3$

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In$\mathrm{arccsc}(-\frac{2\sqrt{3}}{3})$Out$-\frac{π}{3}$

Known value · Wolfram MathematicalFunctionData, ArcCsc

The golden-ratio cosecant gives $3\pi/10$

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In$\mathrm{arccsc}(\sqrt{5}-1)$Out$\frac{3π}{10}$

Known value · Wolfram MathematicalFunctionData, ArcCsc

Negative counterpart gives $-3\pi/10$

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In$\mathrm{arccsc}(1-\sqrt{5})$Out$-\frac{3π}{10}$

Known value · Wolfram MathematicalFunctionData, ArcCsc

$\csc(-\pi/4) = -\sqrt2$

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In$\mathrm{arccsc}(-\sqrt{2})$Out$-\frac{π}{4}$

Known value · Wolfram MathematicalFunctionData, ArcCsc

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In$\mathrm{arccsc}(\sqrt{2-\frac{2\sqrt{5}}{5}})$Out$\frac{2π}{5}$

Known value · Wolfram MathematicalFunctionData, ArcCsc

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In$\mathrm{arccsc}(-\sqrt{2-\frac{2×5^{1/2}}{5}})$Out$-\frac{2π}{5}$

Known value · Wolfram MathematicalFunctionData, ArcCsc

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In$\mathrm{arccsc}(\sqrt{2+\frac{2\sqrt{5}}{5}})$Out$\frac{π}{5}$

Known value · Wolfram MathematicalFunctionData, ArcCsc

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In$\mathrm{arccsc}(-\sqrt{2+\frac{2×5^{1/2}}{5}})$Out$-\frac{π}{5}$

Known value · Wolfram MathematicalFunctionData, ArcCsc

About

Examples	31
Checked against	Wolfram Language 30 agree1 differSageMath 26 agree3 differ1 unevaluatedSymPy 30 agree1 unevaluatedmpmath 25 agree3 errorJulia 24 agree4 errorRust 21 agree6 error
Crosswalk	Wolfram LanguageSageMathSymPympmathJuliaRust

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

asin of the reciprocal, the principal value off 0.

See also: Csc, Arcsin