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Sin

Sin[z]
gives the sine of z.
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • 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 Pi; 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.
EXAMPLESCLOSE ALL
Basic Examples   (4)
The argument is given in radians:
Use Degree to specify an argument in degrees:
The argument is given in radians:
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Use Degree to specify an argument in degrees:
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In[1]:=
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Scope   (10)
Evaluate numerically:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
The precision of the output can be larger than the precision of the input:
Sin threads element-wise over lists and matrices:
Sin can take complex number inputs:
Simple exact values are generated automatically:
More complicated cases require explicit use of FunctionExpand:
Convert multiple-angle expressions:
Convert sums of trigonometric functions to products:
Expand assuming real variables:
Convert to complex exponentials:
TraditionalForm formatting:
Generalizations & Extensions   (4)
Sin can deal with real-valued intervals:
Infinite arguments give symbolic results:
Sin can be applied to power series:
Sin threads element-wise over sparse arrays as well as lists:
Applications   (12)
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:
Triple-periodic surface:
Approximate the almost nowhere differentiable Riemann-Weierstrass function:
Properties & Relations   (10)
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:
Integrals:
Fourier transform:
Sin appears in special cases of many mathematical functions:
Sin is a numeric function:
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:
Use FunctionExpand to express sine of rationals times Pi 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):
Chladni figure:
Plot Sin at integer points:
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