# Sin

Sin[z]

gives the sine of z.

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• Unless explicitly given as a Quantity object, the argument of Sin is assumed to be in radians. (Multiply by Degree 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 can be used with Interval and CenteredInterval objects. »
• Sin automatically threads over lists.

# 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 symbol Degree can be used as a multiplier (e.g. Sin[30 Degree]). When given exact numeric expressions as arguments, Sin may be evaluated to arbitrary numeric precision. Other operations useful for manipulation of symbolic expressions involving Sin include TrigToExp, TrigExpand, Simplify, and FullSimplify.
• 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 by FunctionPeriod. 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 of Sin is ArcSin. The hyperbolic sine is given by Sinh. Other related mathematical functions include Cos, Tan, and Csc.

# Examples

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## Basic Examples(5)

The argument is given in radians:

Use Degree to specify an argument in degrees:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at 0:

## Scope(52)

### Numerical Evaluation(6)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Sin can take complex number inputs:

Evaluate Sin efficiently at high precision:

Sin threads elementwise over lists and matrices:

Sin can be used with Interval and CenteredInterval objects:

### Specific Values(6)

Values of Sin at fixed points:

Sin has exact values at rational multiples of pi:

Values at infinity:

Simple exact values are generated automatically:

More complicated cases require explicit use of FunctionExpand:

Zeros of Sin:

Extrema of Sin:

Find the first positive maximum as a root of :

Substitute in the result:

Visualize the result:

### Visualization(3)

Plot the Sin function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

### 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 is an analytic function of x:

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 for x in [0,π]:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the derivative:

### Integration(3)

Compute the indefinite integral using Integrate:

Definite integral of Sin over a period is 0:

More integrals:

### Series Expansions(4)

Find the Taylor expansion using Series:

Plots of the first three approximations for Sin around :

General term in the series expansion using SeriesCoefficient:

Fourier series:

Sin can be applied to power series:

### Integral Transforms(3)

Compute the Fourier transform using FourierTransform:

### Function Identities and Simplifications(6)

Double-angle formula using TrigExpand:

Angle sum formula:

Multipleangle expressions:

Recover the original expression using TrigReduce:

Convert sums to products using TrigFactor:

Expand using ComplexExpand assuming real variables x and y:

Convert to exponentials using TrigToExp:

### Function Representations(5)

Use Simplify to find a representation through Cos:

Representation through Bessel functions:

Representation through SphericalHarmonicY:

Representation in terms of MeijerG:

Sin can be represented as a DifferentialRoot:

## Applications(15)

Draw a circle:

Lissajous figure:

Equiangular (logarithmic) spiral:

Motion in a circle:

Play a pure tone at 440 Hz:

Solve an equation for harmonic motion:

Rotation matrix:

Rotate a vector:

Plot a sphere:

Plot a torus:

Waves:

Tripleperiodic surface:

Approximate the almost nowhere differentiable RiemannWeierstrass function:

Intensity of the Fraunhofer diffraction pattern of a circular aperture versus diffraction angle:

Encode graphics in a QR code:

Decode and evaluate the expression:

Find a point on a unit circle using Cos and Sin functions:

## Properties & Relations(12)

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:

Fourier transform:

Sin appears in special cases of many mathematical functions:

Sin is a numeric function:

Sin can be represented as a DifferentialRoot:

The generating function for Sin:

The exponential generating function for Sin:

## 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:

Machinenumber inputs can give highprecision results:

Use FunctionExpand to express sine of rationals times using radicals:

Continuous functions involving Sin[x] can give discontinuous indefinite integrals:

In TraditionalForm, 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:

Indefinite integral of :

Plot Sin at integer points:

Wolfram Research (1988), Sin, Wolfram Language function, https://reference.wolfram.com/language/ref/Sin.html (updated 2021).

#### 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_2024_sin, author="Wolfram Research", title="{Sin}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Sin.html}", note=[Accessed: 21-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_sin, organization={Wolfram Research}, title={Sin}, year={2021}, url={https://reference.wolfram.com/language/ref/Sin.html}, note=[Accessed: 21-June-2024 ]}