Sin[z]
gives the sine ofz.
Sin
Sin[z]
gives the sine ofz.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- Unless explicitly given as aQuantity object, the argument ofSin is assumed to be in radians. (Multiply byDegree to convert from degrees.)»
- Sin is automatically evaluated when its argument is a simple rational multiple of
; for more complicated rational multiples,FunctionExpand can sometimes be used.» - For certain special arguments,Sin automatically evaluates to exact values.
- Sin can be evaluated to arbitrary numerical precision.
- Sin automatically threads over lists.»
- Sin can be used withInterval andCenteredInterval objects.»
Background & Context
- Sin is the sine function, which is one of the basic functions encountered in trigonometry. It is defined for real numbers by letting
be a radian angle measured counterclockwise from the
axis along the circumference of the unit circle.Sin[x] then gives the vertical coordinate of the arc endpoint. The equivalent schoolbook definition of the sine of an angle
in a right triangle is the ratio of the length of the leg opposite
to the length of the hypotenuse. - Sin automatically evaluates to exact values when its argument is a simple rational multiple of
. For more complicated rational multiples,FunctionExpand can sometimes be used to obtain an explicit exact value. To specify an argument using an angle measured in degrees, the symbolDegree can be used as a multiplier (e.g.Sin[30Degree]). When given exact numeric expressions as arguments,Sin may be evaluated to arbitrary numeric precision. Other operations useful for manipulation of symbolic expressions involvingSin includeTrigToExp,TrigExpand,Simplify, andFullSimplify. - Sin threads element-wise over lists and matrices. In contrast,MatrixFunction can be used to give the sine of a square matrix (i.e. the power series for the sine function with ordinary powers replaced by matrix powers).
- Sin is periodic with period
, as reported byFunctionPeriod.Sin satisfies the identity
, which is equivalent to the Pythagorean theorem. The definition of the sine function is extended to complex arguments
using the definition
, where
is the base of the natural logarithm. The sine function is entire, meaning it is complex differentiable at all finite points of the complex plane.Sin[z] has series expansion
about the origin. - The inverse function ofSin isArcSin. The hyperbolic sine is given bySinh. Other related mathematical functions includeCos,Tan, andCsc.
Examples
open allclose allBasic Examples (5)
The argument is given in radians:
UseDegree to specify an argument in degrees:
Plot over a subset of the reals:
Scope (52)
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
Sin can take complex number inputs:
EvaluateSin efficiently at high precision:
Compute the elementwise values of an array using automatic threading:
Or compute the matrixSin function usingMatrixFunction:
Compute worst-case guaranteed intervals usingInterval andCenteredInterval objects:
Or compute average-case statistical intervals usingAround:
Specific Values (6)
Values ofSin at fixed points:
Sin has exact values at rational multiples of pi:
Simple exact values are generated automatically:
More complicated cases require explicit use ofFunctionExpand:
Zeros ofSin:
Extrema ofSin:
Visualization (3)
Function Properties (13)
Sin is defined for all real and complex values:
Sin achieves all real values between
and 1:
The range for complex values is the whole plane:
Sin is a periodic function with a period
:
Sin is an odd function:
Sin has the mirror property![sin(TemplateBox[{z}, Conjugate])=TemplateBox[{{sin, (, z, )}}, Conjugate]](/image.pl?url=http%3a%2f%2freference.wolfram.com%2flanguage%2fref%2fFiles%2fSin.en%2f19.png&f=jpg&w=240 "sin(TemplateBox[{z}, Conjugate])=TemplateBox[{{sin, (, z, )}}, Conjugate]")
:
Sin is an analytic function ofx:
Sin is monotonic in a specific range:
Sin is not injective:
Sin is not surjective:
Sin is neither non-negative nor non-positive:
Sin has no singularities or discontinuities:
Sin is neither convex nor concave:
Sin is concave forx in[0,π]:
TraditionalForm formatting:
derivative:Integration (3)
Series Expansions (4)
Find the Taylor expansion usingSeries:
Plots of the first three approximations forSin around
:
General term in the series expansion usingSeriesCoefficient:
Sin can be applied to power series:
Integral Transforms (3)
Function Identities and Simplifications (6)
Double-angle formula usingTrigExpand:
Recover the original expression usingTrigReduce:
Convert sums to products usingTrigFactor:
Expand usingComplexExpand assuming real variablesx andy:
Convert to exponentials usingTrigToExp:
Function Representations (5)
UseSimplify to find a representation throughCos:
Representation through Bessel functions:
Representation throughSphericalHarmonicY:
Representation in terms ofMeijerG:
Sin can be represented as aDifferentialRoot:
Applications (15)
Properties & Relations (13)
Basic parity and periodicity properties are automatically applied:
Complicated expressions containing trigonometric functions do not simplify automatically:
Compose with inverse functions:
Solve a trigonometric equation:
Numerically find a root of a transcendental equation:
Reduce a trigonometric equation:
Sin appears in special cases of many mathematical functions:
Sin is a numeric function:
Sin can be represented as aDifferentialRoot:
The generating function forSin:
The exponential generating function forSin:
Possible Issues (6)
Machine-precision input is insufficient to get a correct answer:
With exact input, the answer is correct:
A larger setting for$MaxExtraPrecision can be needed:
Machine‐number inputs can give high‐precision results:
UseFunctionExpand to express sine of rationals times
using radicals:
Continuous functions involvingSin[x] can give discontinuous indefinite integrals:
InTraditionalForm, parentheses are needed around the argument:
Neat Examples (5)
Noncommensurate waves (quasiperiodic function):
Some arguments can be expressed as a finite sequence of nested radicals:
PlotSin at integer points:
See Also
AngleVector ArcSin Cos Tan Csc Degree SinDegrees TrigToExp TrigExpand Sinc Haversine CirclePoints
Function Repository:SinDegree
History
Introduced in 1988(1.0) |Updated in 1999(4.0)▪2014(10.0)▪2015(10.1)▪2021(13.0)
Text
Wolfram Research (1988), Sin, Wolfram Language function, https://reference.wolfram.com/language/ref/Sin.html (updated 2021).
CMS
Wolfram Language. 1988. "Sin." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Sin.html.
APA
Wolfram Language. (1988). Sin. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Sin.html
BibTeX
@misc{reference.wolfram_2025_sin, author="Wolfram Research", title="{Sin}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Sin.html}", note=[Accessed: 29-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_sin, organization={Wolfram Research}, title={Sin}, year={2021}, url={https://reference.wolfram.com/language/ref/Sin.html}, note=[Accessed: 29-November-2025]}
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