Arcsin

Arcsine, the inverse of Sin restricted to $[-\pi/2, \pi/2]$.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaInverse trigonometric functions↗MathWorldInverseSine↗InverseTrigonometricFunctions↗WikidataQ674533↗nLabinverse trigonometric function↗Britannicatopic/arc-sine↗DLMF4.23.ii.p1↗FungrimAsin✓↗Fungrim entry087a7c✓↗584a61✓↗7954ad↗f0bcb5✓↗Wolfram LanguageArcSin!↗

Arcsin(x)the principal value $y \in [-\pi/2, \pi/2]$ with $\sin(y) = x$.

SageMatharcsin(x)SymPyasin(x)mpmathasin(x)Juliaenumeratio_ball(_asin, x)Rustasin(x)

Typed signatures

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

Domain: Elementary functions

Details

• Real-valued only for $x \in [-1, 1]$; outside that range the result is complex.
• Odd function: $\arcsin(-x) = -\arcsin(x)$.
• Co-function with Arccos: $\arcsin(x) + \arccos(x) = \pi/2$.
• Undoes Sin on its principal branch: $\sin(\arcsin(x)) = x$ for $x \in [-1, 1]$.
• Past $[-1, 1]$, a rational $x$ reduces to the exact closed form $\operatorname{sign}(x)\left(\frac{\pi}{2} - i\ln(|x|+\sqrt{x^2-1})\right)$ (@enumeratio/analytic) -- the branch compute-engine's own N(Arcsin(x)) takes.
• 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. Principal branch, cuts on $(-\infty,-1)$ and $(1,\infty)$: the value on $(1,\infty)$ is the limit from below, so $\arcsin(2) = \frac{\pi}{2} - i\ln(2+\sqrt3)$ (mpmath asin(2)), and oddness gives the other cut.

Examples

Basic #

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In$\arcsin(0)$Out$0$

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In$\arcsin(\frac{1}{2})$Out$\frac{π}{6}$

The top of the principal range

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In$\arcsin(1)$Out$\frac{π}{2}$

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In$\arcsin(-1)$Out$-\frac{π}{2}$

A machine-precision argument evaluates numerically

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In$\arcsin(0.4)$Out$0.411\,516\,846\,067\,488\,019\,385$

Outside $[-1, 1]$ a float argument gives the complex principal value

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In$\arcsin(1.5)$Out$1.570\,796\,326\,794\,896\,6-0.962\,423\,650\,119\,206\,9i$

Scope #

Outside $[-1, 1]$, reduces to the exact closed form $\frac{\pi}{2} - i\ln(2+\sqrt3)$ -- the branch compute-engine's own $N(\mathrm{Arcsin}(2))$ already takes, overriding its plain policy of leaving the call symbolic

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

Arbitrary precision via N(...)

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In$N(\arcsin(\frac{1}{3}))$Out$0.339\,836\,909\,454\,121\,937\,096$

Evaluates at a complex argument

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In$\arcsin(2.5+i)$Out$1.164\,623\,510\,289\,384\,7+1.658\,693\,298\,856\,434\,7i$

Threads element-wise over a matrix

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In$\arcsin(\bigl\lbrack\bigl\lbrack1, \frac{1}{2^{1/2}}\bigr\rbrack, \bigl\lbrack0, \frac{1}{2^{1/2}}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{π}{2}, \frac{π}{4}\bigr\rbrack, \bigl\lbrack0, \frac{π}{4}\bigr\rbrack\bigr\rbrack$

Exact inverse of a special value

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

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

Recognizes the radical value $\sin(\pi/12)$

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In$\arcsin(\frac{\sqrt{6}-\sqrt{2}}{4})$Out$\frac{π}{12}$

Recognizes $\sin(\pi/10) = \frac{\sqrt5 - 1}{4}$

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In$\arcsin(\frac{\sqrt{5}-1}{4})$Out$\frac{π}{10}$

Interval arithmetic: $\arcsin[-\tfrac13, \tfrac12] = [\arcsin(-\tfrac13), \tfrac{\pi}{6}]$

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In$\arcsin(\mathrm{Interval}(-\frac{1}{3}, \frac{1}{2}))$Out$\mathrm{Interval}(\arcsin(-\frac{1}{3}), \frac{π}{6})$

Uncertainty propagation: $\arcsin(0.9 \pm 0.1) \approx 1.120 \pm 0.229$

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In$\arcsin(\mathrm{Around}(0.9, 0.1))$Out$\mathrm{Around}(1.119\,769\,514\,998\,634\,2, 0.229\,415\,733\,870\,561\,8)$

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

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In$\arcsin(-x)$Out$-\arcsin(x)$

Derivative: $\frac{1}{\sqrt{1 - x^2}}$

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

Antiderivative

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In$\int\!\arcsin(x)\, \mathrm{d}x$Out$x\arcsin(x)+\sqrt{1-x^2}$

Taylor series about 0

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In$\operatorname{Series}(\arcsin(x), x, 0, 5)$Out$x+\frac{x^3}{6}+\frac{3x^5}{40}+O\left(x^7\right)$

Already principal: $1/2$ is inside $[-\pi/2, \pi/2]$, so this reduces to itself

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In$\arcsin(\sin(\frac{1}{2}))$Out$\frac{1}{2}$

To 30 significant digits, the last one correctly rounded

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In$N(\arcsin(\frac{1}{3}), 30)$Out$0.339\,836\,909\,454\,121\,937\,096\,392\,513\,392$

$\arcsin(-2)$ past $[-1,1]$: the odd-function mirror of $\arcsin(2)$.

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

Arcsin(+Infinity) is a directed infinity along the negative imaginary axis: $\arcsin(x) \to \pi/2 - i\infty$ as $x \to +\infty$

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Known value · DLMF 4.23.19

Arcsin(-Infinity) is the opposite directed infinity, by oddness: $\arcsin(-x) = -\arcsin(x)$

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Known value · DLMF 4.23.19

Fifty digits of an inverse sine

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In$N(\arcsin(\frac{1}{3}), 50)$Out$0.339\,836\,909\,454\,121\,937\,096\,392\,513\,391\,764\,066\,388\,244\,690\,332\,46$

Every rational past $[-1, 1]$ reduces: $\arcsin(3/2) = \frac{\pi}{2} - i\ln\frac{3+\sqrt5}{2}$

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In$\arcsin(\frac{3}{2})$Out$\frac{π}{2}-i\ln(\frac{3}{2}+\frac{\sqrt{5}}{2})$

Properties #

$\sin(\arcsin(x)) = x$ on the principal branch

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In$\sin(\arcsin(\frac{1}{2}))=\frac{1}{2}$Out$\mathrm{True}$

Co-function identity: $\arcsin(x) + \arccos(x) = \pi/2$

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In$\arcsin(\frac{1}{2})+\arccos(\frac{1}{2})=\frac{π}{2}$Out$\mathrm{True}$

As a logarithm: $\arcsin x = -i\ln(ix + \sqrt{1 - x^2})$

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In$\mathrm{TrigToExp}(\arcsin(x))$Out$-i\ln(i x+\sqrt{1-x^2})$

A nested radical that lands on $-\pi/12$

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In$\arcsin(-\frac{\sqrt{2}}{4}(\sqrt{3}-1))$Out$-\frac{π}{12}$

Known value · Wolfram MathematicalFunctionData, ArcSin

The golden-ratio sine gives $-\pi/10$

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In$\arcsin(\frac{1-\sqrt{5}}{4})$Out$-\frac{π}{10}$

Known value · Wolfram MathematicalFunctionData, ArcSin

A nested radical for $\sin(\pi/8)$

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In$\arcsin(\frac{\sqrt{2-2^{1/2}}}{2})$Out$\frac{π}{8}$

Known value · Wolfram MathematicalFunctionData, ArcSin

Odd symmetry gives $-\pi/8$

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In$\arcsin(-\frac{\sqrt{2-2^{1/2}}}{2})$Out$-\frac{π}{8}$

Known value · Wolfram MathematicalFunctionData, ArcSin

$\sin(-\pi/6) = -\frac12$

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In$\arcsin(-\frac{1}{2})$Out$-\frac{π}{6}$

Known value · Wolfram MathematicalFunctionData, ArcSin

The radical is $\sin(\pi/5)$, recognised in this spelling too

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

Known value · Wolfram MathematicalFunctionData, ArcSin

The negated radical gives $-\pi/5$

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

Known value · Wolfram MathematicalFunctionData, ArcSin

$\sin(-\pi/4) = -\frac{\sqrt2}{2}$

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

Known value · Wolfram MathematicalFunctionData, ArcSin

$\sin(3\pi/10) = \frac{1+\sqrt5}{4}$

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In$\arcsin(\frac{1+\sqrt{5}}{4})$Out$\frac{3π}{10}$

Known value · Wolfram MathematicalFunctionData, ArcSin

Odd symmetry gives $-3\pi/10$

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In$\arcsin(\frac{-5^{1/2}-1}{4})$Out$-\frac{3π}{10}$

Known value · Wolfram MathematicalFunctionData, ArcSin

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

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

Known value · Wolfram MathematicalFunctionData, ArcSin

A nested radical for $\sin(3\pi/8)$

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In$\arcsin(\frac{\sqrt{2+\sqrt{2}}}{2})$Out$\frac{3π}{8}$

Known value · Wolfram MathematicalFunctionData, ArcSin

Possible issues #

Not simply 2: $2$ is outside the principal range $[-\pi/2, \pi/2]$, so $\arcsin(\sin 2) = \pi - 2$

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In$\arcsin(\sin(2))$Out$π-2$

Mirrors through 0: $\arcsin(\sin(-2)) = 2 - \pi$

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In$\arcsin(\sin(-2))$Out$2-π$

Arcsin(ComplexInfinity) is ComplexInfinity: $|\arcsin z| \sim \ln|2z|$ grows in every direction, though the direction itself varies

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Known value · DLMF 4.23.19

About

Examples	45
Checked against	Wolfram Language 42 agree3 differSageMath 39 agree2 differ2 errorSymPy 39 agree4 differmpmath 35 agree2 differJulia 35 agree2 errorRust 30 agree4 error
Crosswalk	WikipediaMathWorld ×2WikidatanLabBritannicaDLMFFungrimFungrim entry ×4Wolfram 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

See also: Sin, Arccos, Arctan