SecDegrees

SecDegrees[θ]

gives the secant of degrees.

Details

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

SecDegrees is related to CosDegrees by the identity .
For certain special arguments, SecDegrees automatically evaluates to exact values.
SecDegrees can be evaluated to arbitrary numerical precision.
SecDegrees automatically threads over lists.
SecDegrees 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 degrees:

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

Calculate the secant by hand:

Verify the result:

Solve a trigonometric equation:

Solve a trigonometric inequality:

Plot over two periods:

Series expansion at 0:

Scope (45)

Numerical Evaluation (5)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate SecDegrees 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 SecDegrees function using MatrixFunction:

Specific Values (6)

Values of SecDegrees at fixed points:

SecDegrees 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 SecDegrees:

Local extrema of SecDegrees:

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

Substitute in the result:

Visualize the result:

Visualization (4)

Plot the SecDegrees function:

Plot over a subset of the complexes:

Plot the real part of SecDegrees:

Plot the imaginary part of SecDegrees:

Polar plot with SecDegrees:

Function Properties (13)

SecDegrees is a periodic function with a period of :

Check this with FunctionPeriod:

The real domain of SecDegrees:

Complex domain:

SecDegrees achieves all real values except the open interval :

The range for complex values:

SecDegrees is an even function:

SecDegrees has the mirror property :

SecDegrees is not an analytic function:

However, it is meromorphic:

SecDegrees is monotonic in a specific range:

SecDegrees is not injective:

SecDegrees is not surjective:

SecDegrees is neither non-negative nor non-positive:

It has both singularity and discontinuity when x is a multiple of 90:

Neither convex nor concave:

It is convex for x in [-90,90]:

TraditionalForm formatting:

Differentiation (3)

First derivative:

Higher derivatives:

Formula for the derivative:

Integration (3)

Compute the indefinite integral of SecDegrees via Integrate:

Definite integral of SecDegrees over a period is 0:

More integrals:

Series Expansions (3)

Find the Taylor expansion using Series:

Plot the first three approximations for SecDegrees around :

Asymptotic expansion at a singular point:

SecDegrees 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 CosDegrees:

Representation through SinDegrees:

Representations through SinDegrees and TanDegrees:

Applications (11)

Basic Trigonomometric Applications (2)

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

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

Trigonomometric Identities (3)

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

Compare with the result of direct calculation:

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 are automatically applied:

Simplify under assumptions on parameters:

Complicated expressions containing trigonometric functions do not simplify automatically:

Another example:

Use FunctionExpand to express SecDegrees in terms of radicals:

Compositions with the inverse trigonometric functions:

Solve a trigonometric equation:

Numerically solve a transcendental equation:

Plot the function to check if the solution is correct:

The zeros of SecDegrees:

The poles of SecDegrees:

Calculate residue symbolically and numerically:

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

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

SecDegrees 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 (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 SecDegrees at integer points:

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