Arcsec

Arcsecant, the inverse of Sec.

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

Arcsec(x)the value $y$ with $\sec(y) = x$.

SageMatharcsec(x)SymPyasec(x)mpmathasec(x)Juliaenumeratio_ball(_asec, x)Rustacos(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 $0<|x|<1$, $\operatorname{arcsec}(x) = \arccos(1/x)$, so $\operatorname{arcsec}(1/2) = i\ln(2+\sqrt3)$ (mpmath asec(0.5)).

Examples

Basic #

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In$\mathrm{arcsec}(1)$Out$0$

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

Scope #

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

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

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In$\mathrm{arcsec}(-1)$Out$π$

$\operatorname{arcsec}(-2) = 2\pi/3$

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

Inside $(-1, 1)$ the value is purely imaginary

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In$\mathrm{arcsec}(0.5)$Out$1.316\,957\,896\,924\,816\,6i$

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In$\mathrm{arcsec}(∞)$Out$\frac{π}{2}$

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

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In$\mathrm{arcsec}(0)$Out$\tilde∞$

Derivative, for real $|x| > 1$

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

Fifty digits of an inverse secant

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In$N(\mathrm{arcsec}(5), 50)$Out$1.369\,438\,406\,004\,565\,827\,776\,196\,139\,422\,128\,031\,858\,546\,618\,285\,3$

Threads over a matrix; zero gives complex infinity

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In$\mathrm{arcsec}(\bigl\lbrack\bigl\lbrack\sqrt{2}, -2^{1/2}\bigr\rbrack, \bigl\lbrack0, -2^{1/2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{π}{4}, \frac{3π}{4}\bigr\rbrack, \bigl\lbrack\tilde∞, \frac{3π}{4}\bigr\rbrack\bigr\rbrack$

$\sec(\pi/6) = \frac{2}{\sqrt3}$

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In$\mathrm{arcsec}(\frac{2\sqrt{3}}{3})$Out$\frac{π}{6}$

At complex infinity $1/z \to 0$, so the value is $\pi/2$

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In$\mathrm{arcsec}(\tilde∞)$Out$\frac{π}{2}$

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

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{arcsec}(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack\sqrt{2}, -2^{1/2}\bigr\rbrack, \bigl\lbrack0, -2^{1/2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{π}{4}, \frac{π}{4}\bigr\rbrack, \bigl\lbrack0, \frac{3π}{4}\bigr\rbrack\bigr\rbrack$

For $0<|x|<1$ the value is $\arccos(1/x)$: $\operatorname{arcsec}(1/2) = i\ln(2+\sqrt3)$

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

And $\operatorname{arcsec}(-1/2) = \pi - i\ln(2+\sqrt3)$

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

Properties #

Undoes Sec: $\sec(\text{arcsec}(x)) = x$

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In$\sec(\mathrm{arcsec}(2))=2$Out$\mathrm{True}$

A difference of radicals giving $\pi/12$

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

Known value · Wolfram MathematicalFunctionData, ArcSec

Swapping the terms gives $11\pi/12$

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In$\mathrm{arcsec}(\sqrt{2}-\sqrt{6})$Out$\frac{11π}{12}$

Known value · Wolfram MathematicalFunctionData, ArcSec

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

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

Known value · Wolfram MathematicalFunctionData, ArcSec

The negated argument stays unevaluated too

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

Known value · Wolfram MathematicalFunctionData, ArcSec

Negative argument lands in $(\pi/2, \pi]$, giving $5\pi/6$

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In$\mathrm{arcsec}(-\frac{2\sqrt{3}}{3})$Out$\frac{5π}{6}$

Known value · Wolfram MathematicalFunctionData, ArcSec

$\sec(\pi/5) = -1 + \sqrt5$

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In$\mathrm{arcsec}(\sqrt{5}-1)$Out$\frac{π}{5}$

Known value · Wolfram MathematicalFunctionData, ArcSec

The negative counterpart gives $4\pi/5$

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In$\mathrm{arcsec}(1-\sqrt{5})$Out$\frac{4π}{5}$

Known value · Wolfram MathematicalFunctionData, ArcSec

$\sec(3\pi/4) = -\sqrt2$

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In$\mathrm{arcsec}(-\sqrt{2})$Out$\frac{3π}{4}$

Known value · Wolfram MathematicalFunctionData, ArcSec

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In$\mathrm{arcsec}(\sqrt{2-\frac{2\sqrt{5}}{5}})$Out$\frac{π}{10}$

Known value · Wolfram MathematicalFunctionData, ArcSec

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In$\mathrm{arcsec}(-\sqrt{2-\frac{2×5^{1/2}}{5}})$Out$\frac{9π}{10}$

Known value · Wolfram MathematicalFunctionData, ArcSec

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In$\mathrm{arcsec}(\sqrt{2+\frac{2\sqrt{5}}{5}})$Out$\frac{3π}{10}$

Known value · Wolfram MathematicalFunctionData, ArcSec

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In$\mathrm{arcsec}(-\sqrt{2+\frac{2×5^{1/2}}{5}})$Out$\frac{7π}{10}$

Known value · Wolfram MathematicalFunctionData, ArcSec

About

Examples	29
Checked against	Wolfram Language 29 agreeSageMath 25 agree2 differ1 unevaluatedSymPy 28 agree1 unevaluatedmpmath 24 agree2 errorJulia 23 agree3 errorRust 20 agree5 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

acos of the reciprocal, the principal value off 0.

See also: Sec, Arccos