TanDegrees

TanDegrees[θ]

gives the tangent of degrees.

Details

TanDegrees and other trigonometric functions are studied in high-school geometry courses and are also used in many scientific disciplines.
The argument of TanDegrees is assumed to be in degrees.
TanDegrees is automatically evaluated when its argument is a simple rational multiple of ; for more complicated rational multiples, FunctionExpand can sometimes be used.
TanDegrees of angle is the ratio of the opposite side to the adjacent side of a right triangle:

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

Calculate the tangent 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:

TanDegrees can take complex number inputs:

Evaluate TanDegrees 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 TanDegrees function using MatrixFunction:

Specific Values (6)

Values of TanDegrees at fixed points:

TanDegrees 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 TanDegrees:

Find one zero using Solve:

Substitute in the result:

Visualize the result:

Singular points of TanDegrees:

Visualization (4)

Plot the TanDegrees function:

Plot over a subset of the complexes:

Plot the real part of TanDegrees:

Plot the imaginary part of TanDegrees:

Polar plot with TanDegrees:

Function Properties (13)

TanDegrees is a periodic function with a period of :

Check this with FunctionPeriod:

Real domain of TanDegrees:

Complex domain:

TanDegrees achieves all real values:

The range for complex values:

TanDegrees is an odd function:

TanDegrees has the mirror property :

TanDegrees is not an analytic function:

However, it is meromorphic:

TanDegrees is monotonic in a specific range:

TanDegrees is not injective:

TanDegrees is surjective:

TanDegrees is neither non-negative nor non-positive:

TanDegrees has both singularities and discontinuities in points multiple to 90:

TanDegrees is neither convex nor concave:

TanDegrees 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 TanDegrees via Integrate:

Definite integral of TanDegrees over a period is 0:

More integrals:

Series Expansions (3)

Find the Taylor expansion using Series:

Plot the first three approximations for TanDegrees around :

Asymptotic expansion at a singular point:

TanDegrees 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 CotDegrees:

Representation through SinDegrees and CosDegrees:

Representation through SecDegrees and CscDegrees:

Applications (12)

Basic Trigonomometric Applications (2)

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

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

Trigonomometric Identities (4)

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

Compare with the result of direct calculation:

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

Compare this result with directly calculated TanDegrees:

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:

The function has the limit zero as approaches :

Thus, its max limit is zero:

Properties & Relations (13)

Check that 1 degree is radians:

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

Simplify under assumptions on parameters:

Complicated expressions containing trigonometric functions do not simplify automatically:

Use FunctionExpand to express TanDegrees 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 TanDegrees:

The poles of TanDegrees:

Calculate residue symbolically and numerically:

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

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

TanDegrees 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 :

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