CotDegrees

CotDegrees[θ]

gives the cotangent of degrees.

Details

  • CotDegrees and other trigonometric functions are studied in high-school geometry courses and are also used in many scientific disciplines.
  • The argument of CotDegrees is assumed to be in degrees.
  • CotDegrees is automatically evaluated when its argument is a simple rational multiple of ; for more complicated rational multiples, FunctionExpand can sometimes be used.
  • CotDegrees of angle is the ratio of the adjacent side to the opposite side of a right triangle:
  • CotDegrees is related to SinDegrees and CosDegrees by the identity TemplateBox[{x}, CotDegrees]=(TemplateBox[{x}, CosDegrees])/(TemplateBox[{x}, SinDegrees]).
  • For certain special arguments, CotDegrees automatically evaluates to exact values.
  • CotDegrees can be evaluated to arbitrary numerical precision.
  • CotDegrees automatically threads over lists.
  • CotDegrees can be used with Interval, CenteredInterval and Around objects.
  • Mathematical function, suitable for both symbolic and numerical manipulation.

Examples

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Basic Examples  (6)

The argument is given in radians:

Calculate CotDegrees of 45 Degree for a right triangle with unit sides:

Calculate the cotangent by hand:

Verify the result:

Solve a trigonometric equation:

Solve a trigonometric inequality:

Plot over two periods:

Series expansion at 0:

Scope  (46)

Numerical Evaluation  (6)

Evaluate numerically:

Evaluate to high precision:

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

CotDegrees can take complex number inputs:

Evaluate CotDegrees efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix CotDegrees function using MatrixFunction:

Specific Values  (6)

Values of CotDegrees at fixed points:

CotDegrees has exact values at rational multiples of 60 degrees:

Values at infinity:

Simple exact values are generated automatically:

More complicated cases require explicit use of FunctionExpand:

Zeros of CotDegrees:

Find one zero using Solve:

Substitute in the result:

Visualize the result:

Singular points of CotDegrees:

Visualization  (4)

Plot the CotDegrees function:

Plot over a subset of the complexes:

Plot the real part of CotDegrees:

Plot the imaginary part of CotDegrees:

Polar plot with CotDegrees:

Function Properties  (13)

CotDegrees is a periodic function with a period of :

Check this with FunctionPeriod:

Real domain of CotDegrees:

Complex domain:

CotDegrees achieves all real values:

The range for complex values:

CotDegrees is an odd function:

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

CotDegrees is not an analytic function:

However, it is meromorphic:

CotDegrees is monotonic in a specific range:

CotDegrees is not injective:

CotDegrees is surjective:

CotDegrees is neither non-negative nor non-positive:

CotDegrees has both singularities and discontinuities in points multiple to 180:

CotDegrees is neither convex nor concave:

CotDegrees is convex for x in [0,90]:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (3)

Compute the indefinite integrals of CotDegrees via Integrate:

Definite integral for CotDegrees over a period:

More integrals:

Series Expansions  (3)

Find the Taylor expansion using Series:

Plot the first three approximations for CotDegrees around :

Asymptotic expansion at a singular point:

CotDegrees can be applied to power series:

Function Identities and Simplifications  (5)

Double-angle formula using TrigExpand:

Angle sum formula:

Multiple‐angle expressions:

Recover the original expression using TrigReduce:

Convert sums to products using TrigFactor:

Convert to exponentials using TrigToExp:

Function Representations  (3)

Representation through TanDegrees:

Representation through SinDegrees and CosDegrees:

Representation through SecDegrees and CscDegrees:

Applications  (12)

Basic Trigonomometric Applications  (2)

Given , find the CotDegrees of the angle using the identity :

Find the missing adjacent side length of a right triangle if the opposite side is 5 and the angle is 30 degrees:

Trigonomometric Identities  (4)

Calculate the CotDegrees value of 105 degrees using the sum and difference formulas:

Compare with the result of direct calculation:

Calculate the CotDegrees value of 15 degrees using the half-angle formula :

Compare this result with directly calculated CotDegrees:

Simplify trigonometric expressions:

Verify trigonometric identities:

Trigonomometric Equations  (2)

Solve a basic trigonometric equation:

Solve trigonometric equations including other trigonometric functions:

Solve trigonometric equations with condition:

Trigonomometric Inequalities  (2)

Solve this trigonometric inequality:

Solve this trigonometric inequality including other trigonometric functions:

Advanced Applications  (2)

Generate a plot over the complex argument plane:

Addition theorem for CotDegrees function:

Properties & Relations  (13)

Check that 1 degree is radians:

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

Simplify with assumptions on parameters:

Complicated expressions containing trigonometric functions do not simplify automatically:

Use FunctionExpand to express CotDegrees in terms of radicals:

Compositions with the inverse trigonometric functions:

Solve a trigonometric equation:

Numerically find a root of a transcendental equation:

Plot the function to check if the solution is correct:

The zeros of CotDegrees:

The poles of CotDegrees:

Calculate residue symbolically and numerically:

FunctionExpand applied to CotDegrees generates expressions in trigonometric functions in radians:

ExpToTrig applied to the outputs of TrigToExp will generate trigonometric functions in radians:

CotDegrees is a numeric function:

Possible Issues  (1)

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

With exact input, the answer is correct:

Neat Examples  (4)

Trigonometric functions are ratios that relate the angle measures of a right triangle to the length of its sides:

Solve trigonometric equations:

Add some condition on the solution:

Some arguments can be expressed as a finite sequence of nested radicals:

Indefinite integral of :

Wolfram Research (2024), CotDegrees, Wolfram Language function, https://reference.wolfram.com/language/ref/CotDegrees.html.

Text

Wolfram Research (2024), CotDegrees, Wolfram Language function, https://reference.wolfram.com/language/ref/CotDegrees.html.

CMS

Wolfram Language. 2024. "CotDegrees." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CotDegrees.html.

APA

Wolfram Language. (2024). CotDegrees. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CotDegrees.html

BibTeX

@misc{reference.wolfram_2024_cotdegrees, author="Wolfram Research", title="{CotDegrees}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/CotDegrees.html}", note=[Accessed: 07-September-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_cotdegrees, organization={Wolfram Research}, title={CotDegrees}, year={2024}, url={https://reference.wolfram.com/language/ref/CotDegrees.html}, note=[Accessed: 07-September-2024 ]}