Arctan

Arctangent, the inverse of Tan restricted to $(-\pi/2, \pi/2)$.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaInverse trigonometric functions↗Arctangent↗MathWorldInverseTangent↗WikidataQ2257242↗nLabarctangent↗DLMF4.23.ii.p1↗FungrimAtan✓↗Fungrim entryWolfram LanguageArcTan!↗

Arctan(x)the principal value $y \in (-\pi/2, \pi/2)$ with $\tan(y) = x$.

SageMatharctan(x)SymPyatan(x)mpmathatan(x)Juliaenumeratio_ball(atan, x)Rustatan(x)

Typed signatures

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

Domain: Elementary functions

Details

• Defined for every real x, unlike Arcsin and Arccos.
• Odd function: $\arctan(-x) = -\arctan(x)$.
• Horizontal asymptotes at $\pm\pi/2$ as $x \to \pm\infty$.
• The two-argument atan2 form is a separate compute-engine function, $\mathrm{Arctan2}(x, y)$, rather than an overload of Arctan.

Examples

Basic #

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In$\arctan(0)$Out$0$

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In$\arctan(1)$Out$\frac{π}{4}$

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In$\arctan(\sqrt{3})$Out$\frac{π}{3}$

A machine-precision argument evaluates numerically

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In$\arctan(0.5)$Out$0.463\,647\,609\,000\,806\,116\,214$

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In$\operatorname{Series}(\arctan(x), x, ∞, 5)$Out$\frac{π}{2}-\frac{1}{x}+\frac{1}{3x^3}-\frac{1}{5x^5}+O\left(\frac{1}{x^7}\right)$

Scope #

Arbitrary precision via N(...)

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In$N(\arctan(\frac{1}{2}))$Out$0.463\,647\,609\,000\,806\,116\,214$

Evaluates at a complex argument

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In$\arctan(2.5+i)$Out$1.233\,425\,855\,683\,120\,4+0.123\,674\,060\,459\,026\,79i$

Wolfram's two-argument ArcTan[x, y] is compute-engine's $\mathrm{Arctan2}(y, x)$: the angle of the point $(-1, -1)$ is $-3\pi/4$

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In$\arctan(-1, -1)$Out$-\frac{3π}{4}$

Arctan2 takes $y$ first: the angle of the point $(0.5, 2)$ is $\arctan 4 \approx 1.3258$, evaluated numerically for the float input

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In$\arctan(2, 0.5)$Out$1.325\,817\,663\,668\,032\,465\,06$

$\arctan(2 + \sqrt3) = 5\pi/12$

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In$\arctan(2+\sqrt{3})$Out$\frac{5π}{12}$

$\arctan(2 - \sqrt3) = \pi/12$

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In$\arctan(2-\sqrt{3})$Out$\frac{π}{12}$

$\arctan(\sqrt2 - 1) = \pi/8$

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In$\arctan(\sqrt{2}-1)$Out$\frac{π}{8}$

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In$\arctan(\frac{1}{\sqrt{3}})$Out$\frac{π}{6}$

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In$\arctan(-1)$Out$-\frac{π}{4}$

Threads element-wise over a matrix

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In$\arctan(\bigl\lbrack\bigl\lbrack0, 1\bigr\rbrack, \bigl\lbrack-1, \sqrt{3}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack0, \frac{π}{4}\bigr\rbrack, \bigl\lbrack-\frac{π}{4}, \frac{π}{3}\bigr\rbrack\bigr\rbrack$

Interval arithmetic: $\arctan[-1, 3] = [-\tfrac{\pi}{4}, \arctan 3]$

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In$\arctan(\mathrm{Interval}(-1, 3))$Out$\mathrm{Interval}(-\frac{π}{4}, \arctan(3))$

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

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

Derivative: $\frac{1}{1 + x^2}$

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In$\frac{\mathrm{d}\arctan(x)}{\mathrm{d}x}$Out$\frac{1}{x^2+1}$

Antiderivative

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In$\int\!\arctan(x)\, \mathrm{d}x$Out$x\arctan(x)-\frac{1}{2}\ln(x^2+1)$

Gregory's series about 0

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In$\operatorname{Series}(\arctan(x), x, 0, 7)$Out$x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+O\left(x^9\right)$

The limit at $+\infty$

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In$\lim_{x→∞}\arctan(x)$Out$\frac{π}{2}$

To 30 significant digits, every one correctly rounded

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In$N(\arctan(\frac{1}{2}), 30)$Out$0.463\,647\,609\,000\,806\,116\,214\,256\,231\,461$

Fifty digits of an inverse tangent

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In$N(\arctan(\frac{1}{2}), 50)$Out$0.463\,647\,609\,000\,806\,116\,214\,256\,231\,461\,214\,402\,028\,537\,054\,286\,12$

Threads over a matrix with exact angles

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In$\arctan(\bigl\lbrack\bigl\lbrack1, \sqrt{3}\bigr\rbrack, \bigl\lbrack0, \sqrt{3}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{π}{4}, \frac{π}{3}\bigr\rbrack, \bigl\lbrack0, \frac{π}{3}\bigr\rbrack\bigr\rbrack$

Properties #

$\tan(\arctan(x)) = x$

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In$\tan(\arctan(1))=1$Out$\mathrm{True}$

As logarithms: $\arctan x = \tfrac{i}{2}\ln(1 - ix) - \tfrac{i}{2}\ln(1 + ix)$

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In$\mathrm{TrigToExp}(\arctan(x))$Out$(-\frac{1}{2}i)\ln(i x+1)+\frac{1}{2}i\ln(1-i x)$

$\tan(\pi/12) = 2-\sqrt3$, so the negation gives $-\pi/12$

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In$\arctan(\sqrt{3}-2)$Out$-\frac{π}{12}$

Known value · Wolfram MathematicalFunctionData, ArcTan

The radical is $\tan(\pi/10)$; with the sign, $-\pi/10$

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

Known value · Wolfram MathematicalFunctionData, ArcTan

$\tan(-\pi/8) = 1-\sqrt2$

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In$\arctan(1-\sqrt{2})$Out$-\frac{π}{8}$

Known value · Wolfram MathematicalFunctionData, ArcTan

$\tan(-\pi/6) = -\frac{1}{\sqrt3}$

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In$\arctan(-\frac{\sqrt{3}}{3})$Out$-\frac{π}{6}$

Known value · Wolfram MathematicalFunctionData, ArcTan

A pentagon tangent giving $\pi/5$

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

Known value · Wolfram MathematicalFunctionData, ArcTan

Odd symmetry gives $-\pi/5$

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

Known value · Wolfram MathematicalFunctionData, ArcTan

The radical is $\tan(3\pi/10)$; with the sign, $-3\pi/10$

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

Known value · Wolfram MathematicalFunctionData, ArcTan

$\tan(-\pi/3) = -\sqrt3$

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

Known value · Wolfram MathematicalFunctionData, ArcTan

$\tan(3\pi/8) = 1+\sqrt2$

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In$\arctan(1+\sqrt{2})$Out$\frac{3π}{8}$

Known value · Wolfram MathematicalFunctionData, ArcTan

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In$N(\arctan(\frac{1}{3})+\arctan(\frac{1}{2}))$Out$0.785\,398\,163\,397\,448\,309\,615$

Known value · Wolfram MathematicalFunctionData, ArcTan, Euler's Machin-like formula

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In$N(2\arctan(\frac{1}{2})-\arctan(\frac{1}{7}))$Out$0.785\,398\,163\,397\,448\,309\,615$

Known value · Wolfram MathematicalFunctionData, ArcTan, Hermann's formula

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In$N(2\arctan(\frac{1}{3})+\arctan(\frac{1}{7}))$Out$0.785\,398\,163\,397\,448\,309\,615$

Known value · Wolfram MathematicalFunctionData, ArcTan, Hutton's formula

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In$N(4\arctan(\frac{1}{5})-\arctan(\frac{1}{239}))$Out$0.785\,398\,163\,397\,448\,309\,615\,47$

Known value · Wolfram MathematicalFunctionData, ArcTan, Machin's formula

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In$\arctan(\sqrt{1-\frac{2\sqrt{5}}{5}})$Out$\frac{π}{10}$

Known value · Wolfram MathematicalFunctionData, ArcTan

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In$\arctan(\sqrt{1+\frac{2\sqrt{5}}{5}})$Out$\frac{3π}{10}$

Known value · Wolfram MathematicalFunctionData, ArcTan

Possible issues #

compute-engine's Arctan takes only one argument and errors on a second; the two-argument atan2 form is the separate $\mathrm{Arctan2}(x, y)$ function

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Arctan(ComplexInfinity) is Indeterminate -- $z = Re^{i\pi/4}$ and $z = -Re^{i\pi/4}$ converge to $+\pi/2$ and $-\pi/2$ as $R \to \infty$

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Known value · DLMF 4.23.26

Neat examples #

The horizontal asymptote as $x \to +\infty$

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In$\arctan(∞)$Out$\frac{π}{2}$

And as $x \to -\infty$

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In$\arctan(-∞)$Out$-\frac{π}{2}$

Euler's Machin-like identity $\arctan\tfrac12 + \arctan\tfrac13 = \tfrac{\pi}{4}$, by FullSimplify; the bare sum stays as written, as in Wolfram

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In$\mathrm{FullSimplify}(\arctan(\frac{1}{2})+\arctan(\frac{1}{3}))$Out$\frac{π}{4}$

Machin's formula $4\arctan\tfrac15 - \arctan\tfrac1{239} = \tfrac{\pi}{4}$, by FullSimplify

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In$\mathrm{FullSimplify}(4\arctan(\frac{1}{5})-\arctan(\frac{1}{239}))$Out$\frac{π}{4}$

About

Examples	47
Checked against	Wolfram Language 45 agree2 differSageMath 42 agree1 unevaluated1 errorSymPy 40 agree2 differmpmath 35 agreeJulia 33 agree2 errorRust 32 agree
Crosswalk	Wikipedia ×2MathWorldWikidatanLabDLMFFungrimFungrim entry ×46Wolfram 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: Tan, Arcsin, Arccos