# 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]:

### 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:

Multipleangle 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:

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 :

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_tandegrees, author="Wolfram Research", title="{TanDegrees}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/TanDegrees.html}", note=[Accessed: 12-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_tandegrees, organization={Wolfram Research}, title={TanDegrees}, year={2024}, url={https://reference.wolfram.com/language/ref/TanDegrees.html}, note=[Accessed: 12-September-2024 ]}