CscDegrees

CscDegrees[θ]

gives the cosecant of degrees.

Details

CscDegrees and other trigonometric functions are studied in high-school geometry courses and are also used in many scientific disciplines.
The argument of CscDegrees is assumed to be in degrees.
CscDegrees of angle is the ratio of the adjacent side to the hypotenuse of a right triangle:

CscDegrees is related to SinDegrees by the identity .
For certain special arguments, CscDegrees automatically evaluates to exact values.
CscDegrees can be evaluated to arbitrary numerical precision.
CscDegrees automatically threads over lists.
CscDegrees 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 CscDegrees of 45 Degree for a right triangle with unit sides:

Calculate the cosecant 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:

Evaluate for complex arguments:

Evaluate CscDegrees 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 CscDegrees function using MatrixFunction:

Specific Values (6)

Values of CscDegrees at fixed points:

CscDegrees has exact values at rational multiples of 30 degrees:

Values at infinity:

Simple exact values are generated automatically:

More complicated cases require explicit use of FunctionExpand:

Singular points of CscDegrees:

Local extrema of CscDegrees:

Find a local minimum of CscDegrees as the root of in the minimum's neighborhood:

Substitute in the result:

Visualize the result:

Visualization (4)

Plot the CscDegrees function:

Plot over a subset of the complexes:

Plot the real part of CscDegrees:

Plot the imaginary part of CscDegrees:

Polar plot with CscDegrees:

Function Properties (13)

CscDegrees is a periodic function with a period of :

Check this with FunctionPeriod:

The real domain of CscDegrees:

Complex domain:

CscDegrees achieves all real values except from the open interval :

The range for complex values:

CscDegrees is an odd function:

CscDegrees has the mirror property :

CscDegrees is not an analytic function:

However, it is meromorphic:

CscDegrees is monotonic in a specific range:

CscDegrees is not injective:

CscDegrees is not surjective:

CscDegrees is neither non-negative nor non-positive:

It has both singularity and discontinuity when x is a multiple of 180:

Neither convex nor concave:

It is convex for x in [0,180]:

TraditionalForm formatting:

Differentiation (3)

First derivative:

Higher derivatives:

Formula for the derivative:

Integration (3)

Compute the indefinite integral of CscDegrees via Integrate:

Definite integral of CscDegrees over a period is 0:

More integrals:

Series Expansions (3)

Find the Taylor expansion using Series:

Plot the first three approximations for CscDegrees around :

Asymptotic expansion at a singular point:

CscDegrees can be applied to power series:

Function Identities and Simplifications (5)

Double-angle formula using TrigExpand:

Angle sum formula:

Multiple‐angle expressions:

Convert sums to products using TrigFactor:

Convert to complex exponentials:

Function Representations (3)

Representation through SinDegrees:

Representation through CosDegrees:

Representations through CosDegrees and CotDegrees:

Applications (11)

Basic Trigonomometric Applications (2)

Given , find the CscDegrees of the angle using the formula :

Find the missing opposite side length of a right triangle with hypotenuse 5 given the angle is 30 degrees:

Trigonomometric Identities (3)

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

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 conditions:

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:

Automatically label different trigonometric functions:

Properties & Relations (13)

Check that 1 degree is radians:

Basic parity and periodicity properties of the cosecant function get automatically applied:

Simplify under assumptions on parameters:

Complicated expressions containing trigonometric functions do not automatically simplify:

Another example:

Use FunctionExpand to express CscDegrees 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 CscDegrees:

The poles of CscDegrees:

Calculate residue symbolically and numerically:

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

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

CscDegrees is a numeric function:

Possible Issues (1)

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

Use arbitrary-precision evaluation instead:

Neat Examples (5)

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 :

Plot CscDegrees at integer points:

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