Cot

Cotangent, the reciprocal of Tan.

✓ checked against that system · ! checked, with disagreements (hover a chip)DLMF4.14.E7↗FungrimCot✓↗Fungrim entryWolfram LanguageCot!↗

Cot(x)the cotangent of x, $\frac{1}{\tan(x)}$.

SageMathcot(x)SymPycot(x)mpmathcot(x)Juliaenumeratio_ball(cot, x)

Typed signatures

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

Domain: Elementary functions

Details

• $\cot(x) = \frac{\cos(x)}{\sin(x)} = \frac{1}{\tan(x)}$.
• Period $\pi$, same as Tan.
• Undefined wherever $\sin(x) = 0$, i.e. at multiples of $\pi$ -- not the same poles as Tan.

Examples

6 more kept as data, checked against the oracles

Basic #

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In$\cot(\frac{π}{6})$Out$\sqrt{3}$

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In$\cot(\frac{π}{4})$Out$1$

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

$\cot 60^\circ = \frac{\sqrt3}{3}$, exactly

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In$\cot(60^{∘})$Out$\frac{\sqrt{3}}{3}$

A machine-precision argument evaluates numerically

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In$\cot(1.2)$Out$0.388\,779\,569\,368\,204\,911\,634$

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In$\operatorname{Series}(\cot(x), x, 0, 10)$Out$\frac{1}{x}-\frac{x}{3}-\frac{x^3}{45}-\frac{2x^5}{945}-\frac{x^7}{4\,725}-\frac{2x^9}{93\,555}+O\left(x^{11}\right)$

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

Scope #

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

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In$N(\cot(\frac{6}{5}))$Out$0.388\,779\,569\,368\,204\,911\,634$

Evaluates at a complex argument

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In$\cot(2.5+i)$Out$-0.275\,669\,121\,229\,899\,45-1.042\,640\,642\,094\,421\,6i$

Threads element-wise over a matrix

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In$\cot(\bigl\lbrack\bigl\lbrack\frac{5π}{6}, \frac{π}{2}\bigr\rbrack, \bigl\lbrack\frac{3π}{2}, -π/6\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack-\sqrt{3}, 0\bigr\rbrack, \bigl\lbrack0, -\sqrt{3}\bigr\rbrack\bigr\rbrack$

$\cot(\pi/8) = 1 + \sqrt2$

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In$\cot(\frac{π}{8})$Out$1+\sqrt{2}$

$\cot(\pi/12) = 2 + \sqrt3$

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In$\cot(\frac{π}{12})$Out$2+\sqrt{3}$

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In$\cot(\frac{π}{5})$Out$\frac{1}{5}\sqrt{5}\sqrt{5+2\sqrt{5}}$

Across the pole at 0 the image comes in two unbounded pieces: $\cot[-\tfrac\pi4, \tfrac\pi4] = (-\infty, -1] \cup [1, \infty)$, a Union of intervals

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In$\cot(\mathrm{Interval}(-\frac{π}{4}, \frac{π}{4}))$Out$\lbrack-∞, -1\rbrack∪\lbrack1, ∞\rbrack$

Uncertainty propagation near a pole grows fast: $\cot(3 \pm 0.01) \approx -7.02 \pm 0.50$

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In$\cot(\mathrm{Around}(3, 0.01))$Out$\mathrm{Around}(-7.015\,252\,551\,434\,534, 0.502\,137\,683\,604\,087\,3)$

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

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

Derivative: $-\csc^2 x$

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In$\frac{\mathrm{d}\cot(x)}{\mathrm{d}x}$Out$-\csc(x)^2$

Antiderivative: $\ln|\sin x|$

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

Fifty digits of a rational argument

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In$N(\cot(\frac{6}{5}), 50)$Out$0.388\,779\,569\,368\,204\,911\,634\,191\,504\,971\,903\,409\,594\,984\,925\,940\,98$

Matrix function giving exact entries

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\cot(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack\frac{5π}{6}, \frac{π}{2}\bigr\rbrack, \bigl\lbrack\frac{3π}{2}, -\frac{π}{6}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{\sqrt{3}}{3}, 0\bigr\rbrack, \bigl\lbrack0, \frac{\sqrt{3}}{3}\bigr\rbrack\bigr\rbrack$

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

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

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

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

Properties #

Reciprocal of Tan: $\cot(x) = \frac{1}{\tan(x)}$

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In$\cot(\frac{π}{4})=\frac{1}{\tan(\frac{π}{4})}$Out$\mathrm{True}$

Pythagorean identity: $1 + \cot^2 x = \csc^2 x$

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In$1+\cot(π/6)^2=\csc(π/6)^2$Out$\mathrm{True}$

$\cot(\pi/10)$ as a nested radical

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In$\cot(\frac{π}{10})$Out$\sqrt{5+2\sqrt{5}}$

Known value · Wolfram MathematicalFunctionData, Cot

$\cot(3\pi/10)$ with the sign flipped under the root

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In$\cot(\frac{3π}{10})$Out$\sqrt{5-2\sqrt{5}}$

Known value · Wolfram MathematicalFunctionData, Cot

$\cot(3\pi/8) = \sqrt2 - 1$

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In$\cot(\frac{3π}{8})$Out$\sqrt{2}-1$

Known value · Wolfram MathematicalFunctionData, Cot

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In$N(\cot(1)\cot(\frac{1}{2}))$Out$1.175\,342\,649\,670\,021\,410\,775\,459\,195\,243\,800\,606\,406\,35$

Known value · Wolfram MathematicalFunctionData, Cot, FunctionalEquations

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In$N(\cot(\frac{1}{2}+\frac{π}{2}))$Out$-0.546\,302\,489\,843\,790\,513\,255$

Known value · Wolfram MathematicalFunctionData, Cot, FunctionalEquations

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In$N(\cot(2)\cot(4))$Out$-0.395\,274\,781\,468\,481\,040\,556\,054\,300\,098\,063\,842\,107\,695$

Known value · Wolfram MathematicalFunctionData, Cot, FunctionalEquations

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In$N(\cot(2+\frac{π}{2}))$Out$2.185\,039\,863\,261\,518\,991\,64$

Known value · Wolfram MathematicalFunctionData, Cot, FunctionalEquations

Possible issues #

A pole: $\sin(0) = 0$

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In$\cot(0)$Out$\tilde∞$

Not a pole here -- Tan's pole at $\pi/2$ is Cot's zero instead

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In$\cot(\frac{π}{2})$Out$0$

Cot(ComplexInfinity) is Indeterminate, for the same recurring poles as Tan

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Known value · DLMF 4.14.10

About

Examples	42
Checked against	Wolfram Language 31 agree10 differ1 unevaluatedSageMath 32 agree1 unevaluated2 errorSymPy 32 agree4 differ1 unevaluatedmpmath 26 agree2 differ1 errorJulia 27 agree2 errorRust 19 agree7 differ1 error
Crosswalk	DLMFFungrimFungrim entry ×7Wolfram LanguageSageMathSymPympmathJulia

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

The reciprocal of tan.

See also: Tan, Sin, Cos