Cos

The cosine of x, in radians.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaSine and cosine↗MathWorldCosine↗WikidataQ13647261↗DLMF4.14.E2↗FungrimCos✓↗Fungrim entryWolfram LanguageCos!↗

Cos(x)the cosine of x, in radians.

SageMathcos(x)!↗SymPycos(x)↗mpmathcos(x)↗Oscarcos(x)Juliaenumeratio_ball(cos, x)Rustcos(x)

Typed signatures

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

Domain: Elementary functions

Details

• Even function: $\cos(-x) = \cos(x)$.
• Period $2\pi$.
• Reciprocal of Sec: $\cos(x) = \frac{1}{\sec(x)}$.
• Co-function with Sin: $\cos(x) = \sin(\pi/2 - x)$.

Examples

7 more kept as data, checked against the oracles

Basic #

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

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

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

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In$\cos(\frac{π}{3})$Out$\frac{1}{2}$

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In$\cos(\frac{π}{2})$Out$0$

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In$\cos(π)$Out$-1$

$\cos 30^\circ$, exactly

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

A machine-precision argument evaluates numerically

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In$\cos(1.2)$Out$0.362\,357\,754\,476\,673\,577\,638$

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In$\operatorname{Series}(\cos(x), x, 0, 10)$Out$1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\frac{x^8}{40\,320}-\frac{x^{10}}{3\,628\,800}+O\left(x^{12}\right)$

Scope #

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

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In$N(\cos(\frac{6}{5}))$Out$0.362\,357\,754\,476\,673\,577\,638$

Evaluates at a complex argument

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In$\cos(2.5+i)$Out$-1.236\,229\,198\,856\,342-0.703\,325\,178\,113\,534\,7i$

Threads element-wise over a matrix

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In$\cos(\bigl\lbrack\bigl\lbrack\frac{5π}{6}, 0\bigr\rbrack, \bigl\lbrack\frac{3π}{2}, -π/2\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack-\frac{3^{1/2}}{2}, 1\bigr\rbrack, \bigl\lbrack0, 0\bigr\rbrack\bigr\rbrack$

$\cos(\pi/12) = \frac{\sqrt6 + \sqrt2}{4}$

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In$\cos(\frac{π}{12})$Out$\frac{\sqrt{2}}{4}+\frac{\sqrt{6}}{4}$

$\cos(\pi/8) = \frac{\sqrt{2+\sqrt2}}{2}$

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

$\cos(\pi/10) = \frac{\sqrt{10+2\sqrt5}}{4}$

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In$\cos(\frac{π}{10})$Out$\frac{1}{4}\sqrt{2}\sqrt{5+\sqrt{5}}$

$\cos(2\pi/5) = \frac{\sqrt5 - 1}{4}$

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

$\cos(3\pi/5) = \frac{1 - \sqrt5}{4}$

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

Reduced into the first quadrant by symmetry

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In$\cos(\frac{11π}{6})$Out$\frac{\sqrt{3}}{2}$

Interval arithmetic: the image of $[\pi/3, \pi/2]$ is $[0, \tfrac12]$

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In$\cos(\mathrm{Interval}(\frac{π}{3}, \frac{π}{2}))$Out$\mathrm{Interval}(0, \frac{1}{2})$

Uncertainty propagation: $\cos(2 \pm 0.01) \approx -0.4161 \pm 0.0091$

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In$\cos(\mathrm{Around}(2, 0.01))$Out$\mathrm{Around}(-0.416\,146\,836\,547\,142\,4, 0.009\,092\,974\,268\,256\,816)$

$\cos(\infty)$ is a type error at boxing, as for Sin; Wolfram gives the oscillation range $[-1, 1]$

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An imaginary argument rewrites to Cosh: $\cos(ix) = \cosh x$

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In$\cos(i x)$Out$\cosh(x)$

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

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

$\cos(\arccos x) = x$ for every $x$

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In$\cos(\arccos(x))$Out$x$

$\cos(\arccos(2)) = 2$, even past $[-1, 1]$

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In$\cos(\arccos(2))$Out$2$

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

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

$\cos(\arctan x) = \frac{1}{\sqrt{1 + x^2}}$

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In$\cos(\arctan(x))$Out$\sqrt{\frac{1}{x^2+1}}$

Derivative: $-\sin x$

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In$\frac{\mathrm{d}\cos(x)}{\mathrm{d}x}$Out$-\sin(x)$

Antiderivative

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In$\int\!\cos(x)\, \mathrm{d}x$Out$\sin(x)$

Taylor series about 0 -- only even powers appear

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In$\operatorname{Series}(\cos(x), x, 0, 6)$Out$1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+O\left(x^8\right)$

To 30 significant digits, every one correctly rounded

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In$N(\cos(1), 30)$Out$0.540\,302\,305\,868\,139\,717\,400\,936\,607\,443$

$10^{100}$ has 101 digits; the exact argument is still reduced modulo $2\pi$ before Cos sees it, not rounded to working precision first

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In$N(\cos(10^{100}))$Out$-0.928\,081\,905\,074\,655\,343\,456$

Known value · Wolfram 15.0.0 Cos

Fifty digits of a rational argument

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In$N(\cos(\frac{6}{5}), 50)$Out$0.362\,357\,754\,476\,673\,577\,638\,373\,355\,623\,076\,020\,339\,947\,785\,576\,65$

Defined everywhere

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In$\mathrm{FunctionDomain}(\cos(x), x)$Out$\mathrm{True}$

The range is $[-1, 1]$

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In$\mathrm{FunctionRange}(\cos(x), x, y)$Out$-1≤y≤1$

The period is $2\pi$

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In$\mathrm{FunctionPeriod}(\cos(x), x)$Out$2π$

Analytic on the whole line

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In$\mathrm{FunctionAnalytic}(\cos(x), x)$Out$\mathrm{True}$

Not injective, since it repeats

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In$\mathrm{FunctionInjective}(\cos(x), x)$Out$\mathrm{False}$

Not surjective onto the reals

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In$\mathrm{FunctionSurjective}(\cos(x), x)$Out$\mathrm{False}$

No singularities

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In$\mathrm{FunctionSingularities}(\cos(x), x)$Out$\mathrm{False}$

No discontinuities

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In$\mathrm{FunctionDiscontinuities}(\cos(x), x)$Out$\mathrm{False}$

A matrix function with exact entries

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\cos(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack\frac{5π}{6}, 0\bigr\rbrack, \bigl\lbrack\frac{3π}{2}, -\frac{π}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack-\frac{3^{1/2}}{2}, 0\bigr\rbrack, \bigl\lbrack-\frac{9\sqrt{3}}{16}, 0\bigr\rbrack\bigr\rbrack$

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

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In$\mathrm{Assuming}(m∈ℤ, \mathrm{FullSimplify}(\mathrm{Refine}(\cos(πm))))$Out$(-1)^{m}$

Applications #

Threads element-wise over a list

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In$\cos(\bigl\lbrack0, \frac{π}{2}, π\bigr\rbrack)$Out$\bigl\lbrack1, 0, -1\bigr\rbrack$

Properties #

Even function: $\cos(-x) = \cos(x)$

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

Co-function identity: $\cos x = \sin(\pi/2 - x)$

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In$\cos(\frac{π}{6})=\sin(\frac{π}{2}-\frac{π}{6})$Out$\mathrm{True}$

Double-angle formula: $\cos 2x = \cos^2 x - \sin^2 x$

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In$\mathrm{TrigExpand}(\cos(2x))$Out$\cos(x)^2-\sin(x)^2$

Angle-addition formula

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In$\mathrm{TrigExpand}(\cos(x+y))$Out$\cos(x)\cos(y)-\sin(x)\sin(y)$

Power reduction: $\cos^2 x = \frac{1 + \cos 2x}{2}$

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In$\mathrm{TrigReduce}(\cos(x)^2)$Out$\frac{\cos(2x)}{2}+\frac{1}{2}$

Euler's formula: $\cos x = \frac{e^{ix} + e^{-ix}}{2}$

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In$\mathrm{TrigToExp}(\cos(x))$Out$\frac{1}{2}(e^{i x}+e^{-i x})$

$\cos(3\pi/10)$ in nested-radical form

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In$\cos(\frac{3π}{10})$Out$\frac{1}{4}\sqrt{2}\sqrt{5-\sqrt{5}}$

Known value · Wolfram MathematicalFunctionData, Cos

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In$\int_{0}^{\frac{π}{2}}\!\cos(t)^2\, \mathrm{d}t$Out$\frac{π}{4}$

Known value · Wolfram MathematicalFunctionData, Cos, Wallis' cosine formula

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In$\prod_{k=1}^2(\frac{5}{4}-\cos(\frac{1}{4}π(2k-1)))$Out$\frac{1}{16}(5-2\sqrt{2})(5+2\sqrt{2})$

Known value · Wolfram MathematicalFunctionData, Cos, Cotes circle property

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In$\int_{0}^{\frac{π}{2}}\!\cos(t)^3\, \mathrm{d}t$Out$\frac{2}{3}$

Known value · Wolfram MathematicalFunctionData, Cos, Wallis' cosine formula

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In$N((i\sin(1/2)+\cos(1/2))^2)$Out$0.540\,302\,305\,868\,139\,717\,401+0.841\,470\,984\,807\,896\,506\,652i$

Known value · Wolfram MathematicalFunctionData, Cos, de Moivre's formula

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In$\prod_{k=1}^3(\frac{5}{4}-\cos(\frac{1}{6}π(2k-1)))$Out$\frac{1}{64}×5(5-2\sqrt{3})(5+2\sqrt{3})$

Known value · Wolfram MathematicalFunctionData, Cos, Cotes circle property

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In$N((i\sin(1/2)+\cos(1/2))^2)$Out$0.540\,302\,305\,868\,139\,717\,401+0.841\,470\,984\,807\,896\,506\,652i$

Known value · Wolfram MathematicalFunctionData, Cos, de Moivre's formula

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In$N((\cos(1/2)-i\sin(1/2))^2)$Out$0.540\,302\,305\,868\,139\,717\,401-0.841\,470\,984\,807\,896\,506\,652i$

Known value · Wolfram MathematicalFunctionData, Cos, de Moivre's formula

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In$N(\int_{-\frac{π}{2}}^{\frac{π}{2}}\!e^{(-(1/2)i)t}\sqrt{\cos(t)}\, \mathrm{d}t)$Out$2.221441469079182±9.992016286406154e-14$

Known value · Wolfram MathematicalFunctionData, Cos, Cauchy's cosine integral formula

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In$N(\int_{0}^{\frac{1}{2}}\!\frac{\cos(t/2)}{\sqrt{\cos(t)-\cos(1/2)}}\, \mathrm{d}t)$Out$2.2214414411509225±0.0007024814642723811$

Known value · Wolfram MathematicalFunctionData, Cos, Mehler-Dirichlet integral

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In$N(-2\cos(1)\cos(1/2)^2+2\cos(1/2)^2+\cos(1)^2)$Out$0.999\,999\,999\,999\,999\,999\,999\,255\,914\,766\,779\,997\,522\,6$

Known value · Wolfram MathematicalFunctionData, Cos, FunctionalEquations

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In$N(\cos(1/2+π/2)^2)$Out$0.229\,848\,847\,065\,930\,141\,299$

Known value · Wolfram MathematicalFunctionData, Cos, FunctionalEquations

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In$N((i\sin(\frac{1}{2})+\cos(\frac{1}{2}))^3)$Out$0.070\,737\,201\,667\,702\,910\,088\,5+0.997\,494\,986\,604\,054\,430\,941i$

Known value · Wolfram MathematicalFunctionData, Cos, de Moivre's formula

Possible issues #

As with Sin, a non-special numeric argument stays symbolic unless it's already a float

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In$\cos(2)$Out$\cos(2)$

$\pi/24$ is past the automatic special-angle table (as in Wolfram), so this stays symbolic

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In$\cos(\frac{π}{24})$Out$\cos(\frac{π}{24})$

Cos(ComplexInfinity) is Indeterminate, for the same unbounded growth off the real axis as Sin

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Known value · DLMF 4.28.9

Neat examples #

$\cos(\pi/5) = \frac{1+\sqrt5}{4}$, related to the golden ratio

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

About

Examples	75
Checked against	Wolfram Language 66 agree9 differSageMath 57 agree1 differ3 errorSymPy 54 agree3 differmpmath 37 agree2 differOscar 16 agreeJulia 39 agree1 errorRust 24 agree10 differ1 error
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimFungrim entry ×59Wolfram LanguageSageMathSymPympmathOscarJuliaRust

Implementation

mappedoscarpackages/oracle/src/mappings.ts

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, Tan, Sec, Arccos