Tan

The tangent of x, in radians.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaTrigonometric functions↗MathWorldTangent↗TrigonometricFunctions↗WikidataQ93344↗nLabtrigonometric function↗Britannicatopic/trigonometric-function↗DLMF4.14.E4↗FungrimTan✓↗Fungrim entry1f026d✓↗1f026d:solve↗c7f7a5↗ee56b9✓↗f516e3↗f516e3:solve↗Wolfram LanguageTan!↗

Tan(x)the tangent of x, in radians.

SageMathtan(x)✓↗SymPytan(x)↗mpmathtan(x)↗Juliaenumeratio_ball(tan, x)Rusttan(x)

Typed signatures

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

Domain: Elementary functions

Details

• $\tan(x) = \frac{\sin(x)}{\cos(x)}$.
• Odd function: $\tan(-x) = -\tan(x)$.
• Period $\pi$ -- half the period of Sin and Cos.
• Undefined wherever $\cos(x) = 0$, i.e. at odd multiples of $\pi/2$; compute-engine returns ComplexInfinity there rather than an error.

Examples

8 more kept as data, checked against the oracles

Basic #

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In$\tan(0)$Out$0$

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

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In$\tan(\frac{π}{4})$Out$1$

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In$\tan(\frac{π}{3})$Out$\sqrt{3}$

$\tan 30^\circ = \frac{\sqrt3}{3}$, exactly

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In$\tan(30^{∘})$Out$\frac{\sqrt{3}}{3}$

A machine-precision argument evaluates numerically

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In$\tan(1.2)$Out$2.572\,151\,622\,126\,318\,935\,41$

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In$\operatorname{Series}(\tan(x), x, 0, 10)$Out$x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{315}+\frac{62x^9}{2\,835}+O\left(x^{11}\right)$

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In$\operatorname{Series}(\tan(x), x, \frac{π}{2}, 5)$Out$-\frac{1}{x-\frac{π}{2}}+\frac{1}{3}(x-\frac{π}{2})+\frac{1}{45}(x-\frac{π}{2})^3+\frac{1}{945}×2(x-\frac{π}{2})^5+O\left((x-\frac{π}{2})^7\right)$

Scope #

N(...) of an exact argument gives an arbitrary-precision value

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In$N(\tan(\frac{6}{5}))$Out$2.572\,151\,622\,126\,318\,935\,41$

Evaluates at a complex argument

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In$\tan(2.5+i)$Out$-0.237\,013\,830\,916\,270\,57+0.896\,437\,917\,127\,861\,7i$

Threads element-wise over a matrix

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In$\tan(\bigl\lbrack\bigl\lbrack\frac{5π}{6}, π\bigr\rbrack, \bigl\lbrack\frac{π}{3}, -π\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack-\frac{3^{1/2}}{3}, 0\bigr\rbrack, \bigl\lbrack\sqrt{3}, 0\bigr\rbrack\bigr\rbrack$

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

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In$\tan(\frac{π}{8})$Out$\sqrt{2}-1$

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

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In$\tan(\frac{π}{12})$Out$2-\sqrt{3}$

$\tan(\pi/5) = \sqrt{5 - 2\sqrt5}$

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

$\tan(\pi/10) = \sqrt{1 - 2/\sqrt5}$

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

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

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

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In$\tan(\frac{3π}{4})$Out$-1$

Interval arithmetic: the image of $[0, \pi/3]$ is $[0, \sqrt3]$

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In$\tan(\mathrm{Interval}(0, \frac{π}{3}))$Out$\mathrm{Interval}(0, \sqrt{3})$

Uncertainty propagation: $\tan(2 \pm 0.02) \approx -2.185 \pm 0.115$

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In$\tan(\mathrm{Around}(2, 0.02))$Out$\mathrm{Around}(-2.185\,039\,863\,261\,519, 0.115\,487\,984\,080\,838\,35)$

$\tan(ix) = i\tanh x$

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In$\tan(i x)$Out$i\tanh(x)$

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

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

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

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

$\tan(\arctan(2)) = 2$

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In$\tan(\arctan(2))$Out$2$

$\tan(\arcsin x) = \frac{x}{\sqrt{1 - x^2}}$

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

Derivative: $\sec^2 x$

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In$\frac{\mathrm{d}\tan(x)}{\mathrm{d}x}$Out$\sec(x)^2$

Antiderivative: $\ln|\sec x|$

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In$\int\!\tan(x)\, \mathrm{d}x$Out$\ln(\vert\sec(x)\vert)$

Taylor series about 0

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In$\operatorname{Series}(\tan(x), x, 0, 7)$Out$x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{315}+O\left(x^9\right)$

To 30 significant digits, every one correctly rounded

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In$N(\tan(1), 30)$Out$1.557\,407\,724\,654\,902\,230\,506\,974\,807\,46$

A 50-digit value of $\tan\frac65$

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In$N(\tan(\frac{6}{5}), 50)$Out$2.572\,151\,622\,126\,318\,935\,409\,994\,236\,033\,363\,956\,529\,409\,306\,043\,4$

Tangent takes every real value

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In$\mathrm{FunctionRange}(\tan(x), x, y)$Out$\mathrm{True}$

The period is $\pi$, not $2\pi$

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In$\mathrm{FunctionPeriod}(\tan(x), x)$Out$π$

Not analytic because of its poles

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In$\mathrm{FunctionAnalytic}(\tan(x), x)$Out$\mathrm{False}$

Meromorphic: analytic except for poles

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In$\mathrm{FunctionMeromorphic}(\tan(x), x)$Out$\mathrm{True}$

Onto the reals

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In$\mathrm{FunctionSurjective}(\tan(x), x)$Out$\mathrm{True}$

Singular where $\cos x=0$

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In$\mathrm{FunctionSingularities}(\tan(x), x)$Out$\cos(x)=0$

Discontinuous exactly at the poles

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In$\mathrm{FunctionDiscontinuities}(\tan(x), x)$Out$\cos(x)=0$

A triangular matrix function with exact entries

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\tan(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack\frac{5π}{6}, π\bigr\rbrack, \bigl\lbrack\frac{π}{3}, -π\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack-\frac{3^{1/2}}{39}, \frac{2\sqrt{3}}{13}\bigr\rbrack, \bigl\lbrack\frac{2\sqrt{3}}{39}, -\frac{4\sqrt{3}}{13}\bigr\rbrack\bigr\rbrack$

$\tan(\infty)$ is a type error at boxing; Wolfram gives the oscillation range, all of $(-\infty, \infty)$

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In$\mathrm{Assuming}(m∈ℤ, \mathrm{Refine}(\tan(πm)))$Out$\tan(πm)$

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In$\mathrm{Assuming}(m∈ℤ, \mathrm{FullSimplify}(\mathrm{Refine}(\tan(π(1/2+m)))))$Out$\tan(π(m+\frac{1}{2}))$

Applications #

Threads element-wise over a list

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In$\tan(\bigl\lbrack0, \frac{π}{4}\bigr\rbrack)$Out$\bigl\lbrack0, 1\bigr\rbrack$

Properties #

$\tan(x) = \frac{\sin(x)}{\cos(x)}$

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In$\tan(\frac{π}{4})=\frac{\sin(\frac{π}{4})}{\cos(\frac{π}{4})}$Out$\mathrm{True}$

Odd function: $\tan(-x) = -\tan(x)$

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In$\tan(-\frac{π}{4})$Out$-1$

Period $\pi$, not $2\pi$: adding a half turn doesn't change the value

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In$\tan(\frac{π}{4}+π)$Out$1$

$\sin x / \cos x$ simplifies to $\tan x$

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In$\mathrm{Simplify}(\frac{\sin(x)}{\cos(x)})$Out$\tan(x)$

Rewritten in exponentials

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In$\mathrm{TrigToExp}(\tan(x))$Out$\frac{-i(e^{i x}-e^{-i x})}{e^{i x}+e^{-i x}}$

$\tan(3\pi/10)$ in nested radicals

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

Known value · Wolfram MathematicalFunctionData, Tan

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

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In$\tan(\frac{3π}{8})$Out$1+\sqrt{2}$

Known value · Wolfram MathematicalFunctionData, Tan

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In$N((1-\tan(1/2)^2)\tan(1))$Out$1.092\,604\,979\,687\,581\,026\,512\,117\,609\,759\,424\,481\,869\,16$

Known value · Wolfram MathematicalFunctionData, Tan, FunctionalEquations

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In$N(\tan(\frac{1}{2}+\frac{π}{2}))$Out$-1.830\,487\,721\,712\,451\,919\,27$

Known value · Wolfram MathematicalFunctionData, Tan, FunctionalEquations

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In$N((1-\tan(2)^2)\tan(4))$Out$-4.370\,079\,726\,523\,037\,983\,281\,860\,374\,993\,791\,294\,936$

Known value · Wolfram MathematicalFunctionData, Tan, FunctionalEquations

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In$N(\tan(2+\frac{π}{2}))$Out$0.457\,657\,554\,360\,285\,763\,75$

Known value · Wolfram MathematicalFunctionData, Tan, FunctionalEquations

Possible issues #

A pole: $\cos(\pi/2) = 0$, so the ratio blows up

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In$\tan(\frac{π}{2})$Out$\tilde∞$

Same rule as Sin and Cos: a non-special numeric argument stays symbolic

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In$\tan(1)$Out$\tan(1)$

Tan(ComplexInfinity) is Indeterminate -- poles recur every period $\pi$, arbitrarily close to any large argument

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Known value · DLMF 4.14.10

About

Examples	63
Checked against	Wolfram Language 54 agree8 differ1 unevaluatedSageMath 47 agree1 unevaluated2 errorSymPy 48 agree2 differ1 unevaluatedmpmath 28 agree2 differ3 unevaluatedJulia 34 agree4 errorRust 27 agree6 differ3 unevaluated
Crosswalk	WikipediaMathWorld ×2WikidatanLabBritannicaDLMFFungrimFungrim entry ×6Wolfram LanguageSageMathSymPympmathJuliaRust

Implementation

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: Sin, Cos, Cot, Arctan