Sin

The sine of x, in radians.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaSine and cosine↗MathWorldSine↗WikidataQ13647261↗DLMF4.14.E1↗FungrimSin✓↗Fungrim entryWolfram LanguageSin!↗

Sin(x)the sine of x, in radians.

SageMathsin(x)✓↗SymPysin(x)↗mpmathsin(x)↗Oscarsin(x)Rustsin(x)

Typed signatures

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

Domain: Elementary functions

Details

• Odd function: $\sin(-x) = -\sin(x)$.
• Period $2\pi$: $\sin(x + 2\pi) = \sin(x)$.
• Reciprocal of Csc: $\sin(x) = \frac{1}{\csc(x)}$.
• compute-engine evaluates exactly at rational multiples of $\pi$ that have a closed radical form -- well past the usual $\pi/6, \pi/4, \pi/3$ set, e.g. $\pi/5$.
• A numeric argument that isn't a recognized special angle, and isn't already floating-point, is left symbolic; pass a float or wrap in N(...) for a numeric approximation.
• Differs from Wolfram: $N(Sin(24^{40}))$ reduces the huge argument exactly and answers correctly; Wolfram's machine-precision N[Sin[24^40]] loses the argument and answers wrongly (it needs an explicit precision).

Examples

6 more kept as data, checked against the oracles

Basic #

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In$\sin(0)$Out$0$

Exact value at a special angle

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In$\sin(\frac{π}{6})$Out$\frac{1}{2}$

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

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

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In$\sin(\frac{π}{2})$Out$1$

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In$\sin(π)$Out$0$

Degrees(d) converts to radians, so $\sin 60^\circ$ is exact

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In$\sin(60^{∘})$Out$\frac{\sqrt{3}}{2}$

A machine-precision argument evaluates numerically

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In$\sin(1.2)$Out$0.932\,039\,085\,967\,226\,349\,67$

Scope #

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

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In$N(\sin(\frac{6}{5}))$Out$0.932\,039\,085\,967\,226\,349\,67$

Evaluates at a complex argument

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In$\sin(2.5+i)$Out$0.923\,490\,776\,043\,173\,1-0.941\,504\,933\,270\,867\,2i$

Threads element-wise over a matrix

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

Exact value at $\pi/12$: $\frac{\sqrt6 - \sqrt2}{4}$

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In$\sin(\frac{π}{12})$Out$\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}$

$\sin(\pi/8) = \frac{\sqrt{2-\sqrt2}}{2}$

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

$\sin(\pi/10) = \frac{\sqrt5 - 1}{4}$

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

$\sin(5\pi/12) = \frac{\sqrt6 + \sqrt2}{4}$

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In$\sin(\frac{5π}{12})$Out$\frac{\sqrt{2}}{4}+\frac{\sqrt{6}}{4}$

$\sin(3\pi/8) = \frac{\sqrt{2+\sqrt2}}{2}$

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

Large rational multiples of $\pi$ are reduced modulo the period first

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In$\sin(\frac{100π}{3})$Out$-\frac{\sqrt{3}}{2}$

Any integer multiple of $\pi$ is a zero of sine, however large

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In$\sin(1\,000\,001π)$Out$0$

$\sin(-\pi) = 0$

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In$\sin(-π)$Out$0$

$\sin(4\pi) = 0$

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In$\sin(4π)$Out$0$

A full-turn shift of a symbolic argument doesn't change the value: $\sin(x + 2\pi) = \sin x$

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In$\sin(x+2π)$Out$\sin(x)$

$\sin(i\pi/2) = i\sinh(\pi/2)$ exactly -- Sinh has no special value at $\pi/2$, so this stays symbolic there

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In$\sin(\frac{i π}{2})$Out$i\sinh(\frac{π}{2})$

N(...) decimalizes $\sin(i\pi/2)$, not just the exact $i\sinh(\pi/2)$ form

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In$N(\sin(\frac{i π}{2}))$Out$2.301\,298\,902\,307\,294\,7i$

Interval arithmetic: the image of $[-\pi/6, \pi/6]$ is $[-\tfrac12, \tfrac12]$

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In$\sin(\mathrm{Interval}(-\frac{π}{6}, \frac{π}{6}))$Out$\mathrm{Interval}(-\frac{1}{2}, \frac{1}{2})$

The image of the whole real line is $[-1, 1]$

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In$\sin(\mathrm{Interval}(-∞, ∞))$Out$\mathrm{Interval}(-1, 1)$

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

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Uncertainty propagation: $\sin(2 \pm 0.01) \approx 0.9093 \pm 0.0042$ to first order

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In$\sin(\mathrm{Around}(2, 0.01))$Out$\mathrm{Around}(0.909\,297\,426\,825\,681\,7, 0.004\,161\,468\,365\,471\,424)$

An imaginary argument rewrites to Sinh: $\sin(ix) = i\sinh x$

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In$\sin(i x)$Out$i\sinh(x)$

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

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

A half-period shift reduces: $\sin(x + \pi) = -\sin x$

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In$\sin(x+π)$Out$-\sin(x)$

$\sin(\arcsin x) = x$ for every $x$

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In$\sin(\arcsin(x))$Out$x$

$\sin(\arcsin(2)) = 2$, even past $[-1, 1]$ where Arcsin's own closed form would otherwise hide the identity from a plain evaluate()

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In$\sin(\arcsin(2))$Out$2$

$\sin(\arcsin(1+2i)) = 1+2i$

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In$\sin(\arcsin(1+2i))$Out$1+2i$

$\sin(\arccos x) = \sqrt{1 - x^2}$

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In$\sin(\arccos(x))$Out$\sqrt{1-x^2}$

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

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

Derivative: $\frac{d}{dx}\sin x = \cos x$

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

Second derivative

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In$\frac{∂^{2}\sin(x)}{∂x^{2}}$Out$-\sin(x)$

Antiderivative

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

Area under one arch: $\int_0^\pi \sin x\,dx = 2$

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In$\int_{0}^{π}\!\sin(x)\, \mathrm{d}x$Out$2$

$\int \sin^3 x\,dx = -\cos x + \tfrac13\cos^3 x$, by $u = \cos x$ -- any odd power of $\sin$ or $\cos$, times any power of the other

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In$\int\!\sin(x)^3\, \mathrm{d}x$Out$\frac{\cos(x)^3}{3}-\cos(x)$

Taylor series about 0 -- only odd powers appear

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In$\operatorname{Series}(\sin(x), x, 0, 10)$Out$x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5\,040}+\frac{x^9}{362\,880}+O\left(x^{11}\right)$

The fundamental limit $\lim_{x\to0} \frac{\sin x}{x} = 1$

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In$\lim_{x→0}\frac{\sin(x)}{x}$Out$1$

To 30 significant digits, every one correctly rounded

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In$N(\sin(1), 30)$Out$0.841\,470\,984\,807\,896\,506\,652\,502\,321\,63$

$24^{40}$ has 55 digits; the exact argument is still reduced modulo $2\pi$ before Sin sees it, not rounded to working precision first

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In$N(\sin(24^{40}))$Out$0.400\,083\,152\,719\,766\,047\,071$

Known value · Wolfram 15.0.0 Sin

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

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In$N(\sin(\frac{6}{5}), 50)$Out$0.932\,039\,085\,967\,226\,349\,670\,134\,435\,494\,825\,995\,415\,070\,588\,208\,73$

Defined for every real $x$

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

The range is $[-1,1]$

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

The period is $2\pi$

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

Entire, so analytic

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

Not injective, since it repeats

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

Not surjective onto the reals

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

No singularities

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

No discontinuities

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

A triangular matrix function giving exact rationals

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In$\mathrm{MatrixFunction}(()↦\sin(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack\frac{5π}{6}, 0\bigr\rbrack, \bigl\lbrack\frac{3π}{2}, -\frac{π}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{1}{2}, 0\bigr\rbrack, \bigl\lbrack\frac{27}{16}, -1\bigr\rbrack\bigr\rbrack$

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In$\mathrm{Assuming}(m∈ℤ, \mathrm{Refine}(\sin(πm)))$Out$0$

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

Applications #

Threads element-wise over a list

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

Degrees(d) converts a degree measure to radians, so familiar degree angles still evaluate exactly

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In$\sin(30^{∘})$Out$\frac{1}{2}$

Properties #

Pythagorean identity $\sin^2 x + \cos^2 x = 1$, here at $x = \pi/6$

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In$\sin(\frac{π}{6})^2+\cos(\frac{π}{6})^2$Out$1$

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

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

Period $2\pi$: adding a full turn doesn't change the value

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In$\sin(\frac{π}{6}+2π)$Out$\frac{1}{2}$

Double-angle formula via TrigExpand

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

Angle-addition formula via TrigExpand

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

Triple-angle formula

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In$\mathrm{TrigExpand}(\sin(3x))$Out$3\sin(x)\cos(x)^2-\sin(x)^3$

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

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

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

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

Simplify knows the Pythagorean identity symbolically

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In$\mathrm{Simplify}(\sin(x)^2+\cos(x)^2)$Out$1$

The same through FullSimplify

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In$\mathrm{FullSimplify}(\sin(x)^2+\cos(x)^2)$Out$1$

$\sin(3\pi/10) = \frac{1+\sqrt5}{4}$

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In$\sin(\frac{3π}{10})$Out$\frac{1}{4}+\frac{\sqrt{5}}{4}$

Known value · Wolfram MathematicalFunctionData, Sin

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

Known value · Wolfram MathematicalFunctionData, Sin, Fresnel integral

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

Known value · Wolfram MathematicalFunctionData, Sin, Wallis' sine formula

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In$\sin(1)^4-2(2\sin(\frac{1}{2})^2-2\sin(\frac{1}{2})^4)\sin(1)^2$Out$\sin(1)^4-4(\sin(1/2)^2-\sin(\frac{1}{2})^4)\sin(1)^2$

Known value · Wolfram MathematicalFunctionData, Sin, FunctionalEquations

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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, Sin, De Moivre's formula

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

Known value · Wolfram MathematicalFunctionData, Sin, Wallis' sine 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, Sin, 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, Sin, De Moivre's formula

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In$N(\int_{0}^{∞}\!\frac{\sin(t)}{t}\, \mathrm{d}t)$Out$1.570\,796\,326\,794\,896\,619\,231\,321\,5$

Known value · Wolfram MathematicalFunctionData, Sin, Dirichlet integral

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In$N(\int_{0}^{∞}\!\frac{\sin(t)}{\sqrt{t}}\, \mathrm{d}t)$Out$1.2533141373155001±1.5505101114093968e-11$

Known value · Wolfram MathematicalFunctionData, Sin, Fresnel integral

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In$N(\int_{0}^{∞}\!\frac{1}{t}(\sin(t/2)\cos(t/2))\, \mathrm{d}t)$Out$0.7853981633974482±6.134909676272713e-12$

Known value · Wolfram MathematicalFunctionData, Sin, Dirichlet discontinuous multiplier

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In$N(\sin(1)^2)$Out$0.708\,073\,418\,273\,571\,193\,5$

Known value · Wolfram MathematicalFunctionData, Sin, FunctionalEquations

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In$N(\sin(1/2+π/2)^2)$Out$0.770\,151\,152\,934\,069\,858\,7$

Known value · Wolfram MathematicalFunctionData, Sin, FunctionalEquations

Possible issues #

1 radian isn't a recognized special angle, so this stays symbolic; wrap in N(...) or pass 1.0 for ≈ 0.841

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In$\sin(1)$Out$\sin(1)$

$\sin(2)$, a generic argument, stays symbolic -- not a recognized special angle

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In$\sin(2)$Out$\sin(2)$

A generic symbolic sum stays symbolic

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In$\sin(x+y)$Out$\sin(x+y)$

$\pi/15$ has a radical form, but like Wolfram's automatic evaluation the table stops short of it -- see FunctionExpand below

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In$\sin(\frac{π}{15})$Out$\sin(\frac{π}{15})$

A huge exact argument is reduced modulo $2\pi$ with enough working precision to get the digits right

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In$N(\sin(10^{20}))$Out$-0.645\,251\,285\,265\,780\,844\,206$

Sin(ComplexInfinity) is Indeterminate -- off the real axis $\sin(iz) = i\sinh(z)$ grows without bound, so it isn't confined to Interval(-1, 1) the way the real infinities are

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Known value · DLMF 4.28.8

Neat examples #

The special-angle table reaches past 30-45-60: $\sin(\pi/5)$ comes out as a nested radical

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

FunctionExpand gives the nested-radical form $\frac{\sqrt{10+2\sqrt5} - \sqrt{15} + \sqrt3}{8}$

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In$\mathrm{FunctionExpand}(\sin(\frac{π}{15}))$Out$\frac{1}{8}(-\sqrt{15}+\sqrt{3}+\sqrt{10+2\sqrt{5}})$

About

Examples	95
Checked against	Wolfram Language 81 agree9 differSageMath 38 agreeSymPy 55 agree2 differmpmath 17 agree1 differOscar 19 agreeRust 26 agree2 differ2 error
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimFungrim entry ×101Wolfram LanguageSageMathSymPympmathOscarRust

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

See also: Cos, Tan, Csc, Arcsin