Sec

Secant, the reciprocal of Cos.

✓ checked against that system · ! checked, with disagreements (hover a chip)DLMF4.14.E6↗FungrimSec↗Wolfram LanguageSec!↗

Sec(x)the secant of x, $\frac{1}{\cos(x)}$.

SageMathsec(x)SymPysec(x)mpmathsec(x)Juliaenumeratio_ball(_sec, x)

Typed signatures

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

Domain: Elementary functions

Details

• $\sec(x) = \frac{1}{\cos(x)}$.
• Even function, period $2\pi$, same as Cos.
• Undefined wherever $\cos(x) = 0$, the same poles as Tan.

Examples

6 more kept as data, checked against the oracles

Basic #

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

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In$\sec(\frac{π}{4})$Out$\sqrt{2}$

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In$\sec(\frac{π}{3})$Out$2$

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In$\sec(π)$Out$-1$

$\sec 30^\circ = \frac{2\sqrt3}{3}$, exactly

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In$\sec(30^{∘})$Out$\frac{2\sqrt{3}}{3}$

A machine-precision argument evaluates numerically

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In$\sec(1.2)$Out$2.759\,703\,601\,332\,406\,456\,89$

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In$\operatorname{Series}(\sec(x), x, 0, 10)$Out$1+\frac{x^2}{2}+\frac{5x^4}{24}+\frac{61x^6}{720}+\frac{277x^8}{8\,064}+\frac{50\,521x^{10}}{3\,628\,800}+O\left(x^{12}\right)$

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In$\operatorname{Series}(\sec(x), x, \frac{3π}{2}, 5)$Out$\frac{1}{x-\frac{3π}{2}}+\frac{1}{6}(x-\frac{3π}{2})+\frac{1}{360}×7(x-\frac{3π}{2})^3+\frac{1}{15\,120}×31(x-\frac{3π}{2})^5+O\left((x-\frac{3π}{2})^7\right)$

Scope #

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

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In$N(\sec(\frac{6}{5}))$Out$2.759\,703\,601\,332\,406\,456\,89$

Evaluates at a complex argument

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In$\sec(2.5+i)$Out$-0.611\,108\,564\,155\,231\,6+0.347\,676\,660\,710\,495\,97i$

Threads element-wise over a matrix

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In$\sec(\bigl\lbrack\bigl\lbrack\frac{5π}{6}, 0\bigr\rbrack, \bigl\lbrack\frac{π}{6}, -π/4\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack-\frac{2\sqrt{3}}{3}, 1\bigr\rbrack, \bigl\lbrack\frac{2\sqrt{3}}{3}, \sqrt{2}\bigr\rbrack\bigr\rbrack$

$\sec(\pi/12) = \sqrt6 - \sqrt2$

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

$\sec(\pi/5) = \sqrt5 - 1$

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In$\sec(\frac{π}{5})$Out$\sqrt{5}-1$

$\sec(\pi/8) = \sqrt{4 - 2\sqrt2}$

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In$\sec(\frac{π}{8})$Out$\sqrt{4-2\sqrt{2}}$

Interval arithmetic: the image of $[\pi/6, \pi/3]$ is $[\tfrac{2}{\sqrt3}, 2]$

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In$\sec(\mathrm{Interval}(\frac{π}{6}, \frac{π}{3}))$Out$\mathrm{Interval}(\frac{2\sqrt{3}}{3}, 2)$

Uncertainty propagation: $\sec(2 \pm 0.01) \approx -2.403 \pm 0.053$

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In$\sec(\mathrm{Around}(2, 0.01))$Out$\mathrm{Around}(-2.402\,997\,961\,722\,381, 0.052\,506\,463\,376\,995\,806)$

Evenness drops the sign of a symbolic argument: $\sec(-x) = \sec x$

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

Derivative: $\sec x \tan x$

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

Antiderivative: $\ln|\sec x + \tan x|$

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

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

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In$N(\sec(\frac{6}{5}), 50)$Out$2.759\,703\,601\,332\,406\,456\,883\,432\,939\,237\,783\,860\,073\,115\,323\,011$

A triangular matrix with exact entries from eigenvalues $\frac{5\pi}6$ and $-\frac\pi4$

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\sec(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack\frac{5π}{6}, 0\bigr\rbrack, \bigl\lbrack\frac{π}{6}, -\frac{π}{4}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack-\frac{2\sqrt{3}}{3}, 0\bigr\rbrack, \bigl\lbrack-\frac{2\sqrt{2}}{13}-\frac{4\sqrt{3}}{39}, \sqrt{2}\bigr\rbrack\bigr\rbrack$

$\sec(\infty)$ is a type error at boxing; Wolfram gives the oscillation range, $(-\infty, -1] \cup [1, \infty)$

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

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

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

Properties #

Reciprocal of Cos: $\sec(x) = \frac{1}{\cos(x)}$

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In$\sec(\frac{π}{3})=\frac{1}{\cos(\frac{π}{3})}$Out$\mathrm{True}$

Pythagorean identity: $1 + \tan^2 x = \sec^2 x$

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

$\sec(\pi/10)$ in nested radicals

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

Known value · Wolfram MathematicalFunctionData, Sec

$\sec(3\pi/10)$, the conjugate nested radical

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

Known value · Wolfram MathematicalFunctionData, Sec

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In$N((2-\sec(1/2)^2)\sec(1))$Out$1.298\,446\,410\,409\,524\,836\,889\,507\,601\,337\,994\,043\,479\,2$

Known value · Wolfram MathematicalFunctionData, Sec, FunctionalEquations

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In$N(2\sec(1)\sec(1/2)^2+(\sec(\frac{1}{2})^4-2\sec(1/2)^2)\sec(1)^2-\sec(\frac{1}{2})^4)$Out$-4.654\,804\,648\,704\,176\,015\,24\cdot10^{-20}$

Known value · Wolfram MathematicalFunctionData, Sec, FunctionalEquations

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In$N(\sec(1/2+π/2)^2)$Out$4.350\,685\,299\,340\,042\,821\,54$

Known value · Wolfram MathematicalFunctionData, Sec, FunctionalEquations

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In$N((2-\sec(2)^2)\sec(4))$Out$5.774\,399\,204\,041\,917\,612\,379\,781\,745\,112\,976\,437\,665\,7$

Known value · Wolfram MathematicalFunctionData, Sec, FunctionalEquations

Possible issues #

A pole: $\cos(\pi/2) = 0$

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In$\sec(\frac{π}{2})$Out$\tilde∞$

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

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Known value · DLMF 4.14.10

About

Examples	41
Checked against	Wolfram Language 31 agree9 differ1 unevaluatedSageMath 31 agree1 unevaluated2 errorSymPy 33 agree3 differmpmath 25 agree2 differ1 unevaluatedJulia 26 agree2 errorRust 19 agree6 differ1 unevaluated
Crosswalk	DLMFFungrimWolfram 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 cos.

See also: Cos, Csc, Cot