Cot

Cot[z]

gives the cotangent of z.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The argument of Cot is assumed to be in radians. (Multiply by Degree to convert from degrees.)
  • .
  • Cos[z]/Sin[z] is automatically converted to Cot[z]. TrigFactorList[expr] does decomposition.
  • For certain special arguments, Cot automatically evaluates to exact values.
  • Cot can be evaluated to arbitrary numerical precision.
  • Cot automatically threads over lists.

Background & Context

  • Cot is the cotangent function, which is one of the basic functions encountered in trigonometry. It is defined as the reciprocal of the tangent function: . The equivalent schoolbook definition of the cotangent of an angle in a right triangle is the ratio of the length of the leg adjacent to to the length of the leg opposite it.
  • Cot 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. TrigFactorList can be used to factor expressions involving Cot into terms containing Sin and Cos. To specify an argument using an angle measured in degrees, the symbol Degree can be used as a multiplier (e.g. Cot[30 Degree]). When given exact numeric expressions as arguments, Cot may be evaluated to arbitrary numeric precision. Other operations useful for manipulation of symbolic expressions involving Cot include TrigToExp, TrigExpand, Simplify, and FullSimplify.
  • Cot threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the cotangent of a square matrix (i.e. the power series for the cotangent function with ordinary powers replaced by matrix powers) as opposed to the cotangents of the individual matrix elements.
  • Cot is periodic with period , as reported by FunctionPeriod. Cot satisfies the identity , which is equivalent to the Pythagorean theorem. The definition of the cotangent function is extended to complex arguments using the definition , where is the base of the natural logarithm. Cot has poles at for an integer and evaluates to ComplexInfinity at these points. Cot[z] has series expansion sum_(k=0)^infty((-1)^k 2^(2 k) TemplateBox[{{2,  , k}}, BernoulliB] )/((2 k)!)z^(2 k-1) about the origin that may be expressed in terms of the Bernoulli numbers BernoulliB.
  • The inverse function of Cot is ArcCot. The hyperbolic cotangent is given by Coth. Other related mathematical functions include Tan and Cos.

Examples

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Basic Examples  (6)

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:

Asymptotic expansion at a singular point:

Scope  (39)

Numerical Evaluation  (6)

Evaluate numerically:

Evaluate to high precision:

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

Cot can take complex number inputs:

Evaluate Cot efficiently at high precision:

Cot can deal with realvalued intervals:

Cot threads elementwise over lists and matrices:

Specific Values  (5)

Values of Cot at fixed points:

Values at infinity:

Zeros of Cot:

Find a zero of Cot using Solve:

Substitute in the result:

Visualize the result:

Singular points of Cot:

Simple exact values are generated automatically:

More complicated cases require explicit use of FunctionExpand:

Visualization  (3)

Plot the Cot function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

Function Properties  (6)

The real domain of Cot:

Complex domain:

Cot achieves all real values:

Cot is a periodic function with a period :

Cot is an odd function:

Cot has the mirror property cot(TemplateBox[{z}, Conjugate])=TemplateBox[{{cot, (, z, )}}, Conjugate]:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of Cot:

Definite integral for Cot over a period:

More integrals:

Series Expansions  (3)

Find the Taylor expansion using Series:

Plot the first three approximations for Cot around :

General term in the series expansion of Cot:

Cot can be applied to power series:

Function Identities and Simplifications  (6)

Cot of a double angle:

Cot of a sum:

Convert multipleangle expressions:

Convert sums of trigonometric functions to products:

Expand assuming real variables and :

Convert to complex exponentials:

Function Representations  (4)

Representation through Tan:

Representation through Jacobi functions:

Representation through SphericalHarmonicY:

Representation through Mathieu functions:

Applications  (3)

Generate a plot with poles removed:

Generate a plot over the complex argument plane:

The cotangent function conformally maps a parabola into the unit disk:

Properties & Relations  (11)

Basic parity and periodicity properties of the cotangent function are automatically applied:

Use TrigFactorList to factor Cot into Sin and Cos:

Complicated expressions containing trigonometric functions do not simplify automatically:

Simplify with assumptions on parameters:

Compose with inverse functions:

Solve a trigonometric equation:

Solve for zeros and poles:

Numerically find a root of a transcendental equation:

Cot appears in special cases of many mathematical functions:

Calculate residue symbolically and numerically:

Cot is a numeric function:

Possible Issues  (3)

Machine-precision input is insufficient to give a correct answer:

With exact input, the answer is correct:

A larger setting for $MaxExtraPrecision is needed:

The precision of the output can be much smaller than the precision of the input:

Neat Examples  (6)

Plot Cot at integer points:

The continued fraction is highly regular:

Introduced in 1988
 (1.0)
 |
Updated in 1996
 (3.0)