gives the cotangent of z.
- 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 can be used with Interval and CenteredInterval objects. »
- 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 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.
Examplesopen allclose all
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:
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
Cot can take complex number inputs:
Evaluate Cot efficiently at high precision:
Cot threads elementwise over lists and matrices:
Cot can be used with Interval and CenteredInterval objects:
Specific Values (5)
Plot the Cot function:
Function Properties (13)
The real domain of Cot:
Cot achieves all real values:
Cot is a periodic function with a period :
Cot is an odd function:
Cot has the mirror property :
Cot is not an analytic function:
Cot is monotonic in a specific range:
Cot is not injective:
Cot is surjective:
Cot is neither non-negative nor non-positive:
Cot has both singularities and discontinuities in points multiple to π:
Cot is neither convex nor concave:
Series Expansions (3)
Function Identities and Simplifications (6)
Function Representations (4)
Representation through Tan:
Representation through Jacobi functions:
Representation through SphericalHarmonicY:
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:
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:
Wolfram Research (1988), Cot, Wolfram Language function, https://reference.wolfram.com/language/ref/Cot.html (updated 2021).
Wolfram Language. 1988. "Cot." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Cot.html.
Wolfram Language. (1988). Cot. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Cot.html