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 .
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

open allclose all

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 :

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 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 :

Top