Arccos

Arccosine, the inverse of Cos restricted to $[0, \pi]$.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaInverse trigonometric functions↗MathWorldInverseCosine↗InverseTrigonometricFunctions↗WikidataQ674533↗nLabinverse trigonometric function↗Britannicatopic/arc-sine↗DLMF4.23.ii.p1↗FungrimAcos✓↗Fungrim entry33e034✓↗718f3a✓↗b8fdcd✓↗ec7f2d✓↗fda800✓↗Wolfram LanguageArcCos!↗

Arccos(x)the principal value $y \in [0, \pi]$ with $\cos(y) = x$.

SageMatharccos(x)SymPyacos(x)mpmathacos(x)Juliaenumeratio_ball(_acos, x)Rustacos(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.
• Co-function with Arcsin: $\arccos(x) = \pi/2 - \arcsin(x)$.
• Undoes Cos on its principal branch: $\cos(\arccos(x)) = x$ for $x \in [-1, 1]$.
• 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, same cuts as Arcsin: for $x>1$, $\arccos(x) = i\ln(x+\sqrt{x^2-1})$ (mpmath acos(2) is $1.317i$), and for $x<-1$, $\pi - i\ln(|x|+\sqrt{x^2-1})$.

Examples

1 more kept as data, checked against the oracles

Basic #

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In$\arccos(1)$Out$0$

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In$\arccos(\frac{1}{2})$Out$\frac{π}{3}$

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In$\arccos(0)$Out$\frac{π}{2}$

The top of the principal range

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In$\arccos(-1)$Out$π$

A machine-precision argument evaluates numerically

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In$\arccos(0.4)$Out$1.159\,279\,480\,727\,408\,599\,85$

Outside $[-1, 1]$ the value is purely imaginary

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In$\arccos(1.5)$Out$0.962\,423\,650\,119\,206\,9i$

Scope #

Arbitrary precision via N(...)

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In$N(\arccos(\frac{1}{3}))$Out$1.230\,959\,417\,340\,774\,682\,14$

Evaluates at a complex argument

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In$\arccos(2.5+i)$Out$0.406\,172\,816\,505\,511\,85-1.658\,693\,298\,856\,434\,7i$

Threads element-wise over a matrix

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

Exact inverse of a special value

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

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

Recognizes $\cos(\pi/5) = \frac{1 + \sqrt5}{4}$

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In$\arccos(\frac{1+\sqrt{5}}{4})$Out$\frac{π}{5}$

Interval arithmetic: decreasing, so $\arccos[\tfrac13, \tfrac12] = [\tfrac{\pi}{3}, \arccos\tfrac13]$

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

Uncertainty propagation: $\arccos(0.9 \pm 0.1) \approx 0.451 \pm 0.229$

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In$\arccos(\mathrm{Around}(0.9, 0.1))$Out$\mathrm{Around}(0.451\,026\,811\,796\,262\,36, 0.229\,415\,733\,870\,561\,8)$

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

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

Antiderivative

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

Not simply 4: $4$ is outside the principal range $[0, \pi]$, so $\arccos(\cos 4) = 2\pi - 4$

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In$\arccos(\cos(4))$Out$2π-4$

Arccos(+Infinity) is a directed infinity along the positive imaginary axis: $\arccos = \pi/2 - \arcsin$

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Known value · DLMF 4.23.22

Arccos(-Infinity) is the opposite directed infinity

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Known value · DLMF 4.23.22

Fifty digits of an algebraic argument

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In$N(\arccos(\frac{1}{3}), 50)$Out$1.230\,959\,417\,340\,774\,682\,134\,929\,178\,247\,987\,375\,710\,340\,009\,355\,1$

The integral of $\arccos$ over $[-1,1]$ is exactly $\pi$

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In$\int_{-1}^{1}\!\arccos(x)\, \mathrm{d}x$Out$π$

Past $[-1, 1]$ the same reduction as Arcsin: $\arccos(2) = i\ln(2+\sqrt3)$, the co-function $\frac{\pi}{2} - \arcsin(2)$

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In$\arccos(2)$Out$i\ln(2+\sqrt{3})$

The mirror past $-1$: $\arccos(-2) = \pi - i\ln(2+\sqrt3)$

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

Properties #

$\cos(\arccos(x)) = x$ on the principal branch

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In$\cos(\arccos(\frac{1}{2}))=\frac{1}{2}$Out$\mathrm{True}$

Reflection: $\arccos(-x) = \pi - \arccos x$

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In$\arccos(-x)$Out$π-\arccos(x)$

A nested radical that lands on $5\pi/12$

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In$\arccos(\frac{1}{4}\sqrt{2}(\sqrt{3}-1))$Out$\frac{5π}{12}$

Known value · Wolfram MathematicalFunctionData, ArcCos

Negating the argument gives $\pi$ minus the previous angle

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In$\arccos(-\frac{\sqrt{2}}{4}(\sqrt{3}-1))$Out$\frac{7π}{12}$

Known value · Wolfram MathematicalFunctionData, ArcCos

The golden-ratio cosine gives $2\pi/5$

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In$\arccos(\frac{\sqrt{5}-1}{4})$Out$\frac{2π}{5}$

Known value · Wolfram MathematicalFunctionData, ArcCos

Negative golden-ratio cosine gives $3\pi/5$

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In$\arccos(\frac{1-\sqrt{5}}{4})$Out$\frac{3π}{5}$

Known value · Wolfram MathematicalFunctionData, ArcCos

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

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In$\arccos(\frac{\sqrt{2-2^{1/2}}}{2})$Out$\frac{3π}{8}$

Known value · Wolfram MathematicalFunctionData, ArcCos

Negation reflects it to $5\pi/8$

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In$\arccos(-\frac{\sqrt{2-2^{1/2}}}{2})$Out$\frac{5π}{8}$

Known value · Wolfram MathematicalFunctionData, ArcCos

$\cos(2\pi/3) = -\frac12$

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In$\arccos(-\frac{1}{2})$Out$\frac{2π}{3}$

Known value · Wolfram MathematicalFunctionData, ArcCos

The nested radical is $\cos(3\pi/10)$

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

Known value · Wolfram MathematicalFunctionData, ArcCos

Its negative is $\cos(7\pi/10)$

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In$\arccos(-\frac{1}{2}\sqrt{\frac{5-5^{1/2}}{2}})$Out$\frac{7π}{10}$

Known value · Wolfram MathematicalFunctionData, ArcCos

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

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

Known value · Wolfram MathematicalFunctionData, ArcCos

Cosine of $4\pi/5$ is a negative golden-ratio value

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

Known value · Wolfram MathematicalFunctionData, ArcCos

$\cos(5\pi/6) = -\frac{\sqrt3}{2}$

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

Known value · Wolfram MathematicalFunctionData, ArcCos

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

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

Known value · Wolfram MathematicalFunctionData, ArcCos

Possible issues #

Arccos(ComplexInfinity) is ComplexInfinity, unbounded in every direction as Arcsin is

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Known value · DLMF 4.23.22

About

Examples	40
Checked against	Wolfram Language 37 agree3 differSageMath 34 agree2 differ2 errorSymPy 35 agree4 differmpmath 31 agree2 differJulia 30 agree2 errorRust 27 agree3 error
Crosswalk	WikipediaMathWorld ×2WikidatanLabBritannicaDLMFFungrimFungrim entry ×5Wolfram 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: Cos, Arcsin, Arctan