Introduction to the Cosine Function in Mathematica

Overview

The following shows how the cosine function is realized in Mathematica. Examples of evaluating Mathematica functions applied to various numeric and exact expressions that involve the cosine function or return it are shown. These involve numeric and symbolic calculations and plots.

Notations

Mathematica forms of notations

Following Mathematica's general naming convention, function names in StandardForm are just the capitalized versions of their traditional mathematics names. This shows the cosine function in StandardForm.

This shows the cosine function in TraditionalForm.

Additional forms of notations

Mathematica also knows the most popular forms of notations for the cosine function that are used in other programming languages. Here are three examples: CForm, TeXForm, and FortranForm.

Automatic evaluations and transformations

Evaluation for exact, machine-number, and high-precision arguments

For the exact argument , Mathematica returns an exact result.

For a machine‐number argument (a numerical argument with a decimal point and not too many digits), a machine number is also returned.

The next inputs calculate 100‐digit approximations at and .

Within a second, it is possible to calculate thousands of digits for the cosine function. The next input calculates 10000 digits for and analyzes the frequency of the digit in the resulting decimal number.

Here is a 50‐digit approximation of the cosine function at the complex argument .

Mathematica automatically evaluates mathematical functions with machine precision, if the arguments of the function are machine‐number elements. In this case, only six digits after the decimal point are shown in the results. The remaining digits are suppressed, but can be displayed using the function InputForm.

Simplification of the argument

Mathematica knows the symmetry and periodicity of the cosine function. Here are some examples:

Mathematica automatically simplifies the composition of the direct and the inverse cosine functions into its argument.

Mathematica also automatically simplifies the composition of the direct and any of the inverse trigonometric functions into algebraic functions of the argument.

If the argument has the structure or , and or with integer , the cosine function can be automatically transformed into trigonometric or hyperbolic sine or cosine functions.

Simplification of simple expressions containing the cosine function

Sometimes simple arithmetic operations containing the cosine function can automatically produce other trigonometric functions.

The cosine function arising as special cases from more general functions

The cosine function can be treated as a particular case of some more general special functions. For example, can appear automatically from Bessel, Mathieu, Jacobi, hypergeometric, and Meijer functions for appropriate values of their parameters.

Equivalence transformations carried out by specialized Mathematica functions

General remarks

Almost everybody prefers using instead of . Mathematica automatically transforms the second expression into the first one. The automatic application of transformation rules to mathematical expressions can give overly complicated results. Compact expressions like should not be automatically expanded into the more complicated expression . Mathematica has special functions that produce such expansions. Some are demonstrated in the next section.

TrigExpand

The function TrigExpand expands out trigonometric and hyperbolic functions. In more detail, it splits up sums and integer multiples that appear in arguments of trigonometric and hyperbolic functions, and then expands out products of trigonometric and hyperbolic functions into sums of powers, using trigonometric and hyperbolic identities where possible. Here are some examples.

TrigFactor

The function TrigFactor factors trigonometric and hyperbolic functions. In more detail, it splits up sums and integer multiples that appear in the arguments of trigonometric and hyperbolic functions, and then factors the resulting polynomials into trigonometric and hyperbolic functions, using trigonometric and hyperbolic identities where possible. Here are some examples.

TrigReduce

The function TrigReduce rewrites the products and powers of trigonometric and hyperbolic functions in terms of trigonometric and hyperbolic functions with combined arguments. In more detail, it typically yields a linear expression involving trigonometric and hyperbolic functions with more complicated arguments. TrigReduce is approximately inverse to TrigExpand and TrigFactor. Here are some examples.

TrigToExp

The function TrigToExp converts trigonometric and hyperbolic functions to exponentials. It tries, where possible, to give results that do not involve explicit complex numbers. Here are some examples.

ExpToTrig

The function ExpToTrig converts exponentials to trigonometric and hyperbolic functions. It is approximately inverse to TrigToExp. Here are some examples.

ComplexExpand

The function ComplexExpand expands expressions assuming that all the variables are real. The value option TargetFunctions is a list of functions from the set {Re, Im, Abs, Arg, Conjugate, Sign}. ComplexExpand tries to give results in terms of the functions specified. Here are some examples.

Simplify

The function Simplify performs a sequence of algebraic transformations on its argument, and returns the simplest form it finds. Here are some examples.

Here is a large collection of trigonometric identities. All are written as one large logical conjunction.

The function Simplify has the Assumption option. For example, Mathematica knows that for all real , and knows about the periodicity of trigonometric functions for the symbolic integer coefficient of .

Mathematica also knows that the composition of the inverse and direct trigonometric functions produces the value of the internal argument under the corresponding restriction.

FunctionExpand (and Together)

While the cosine function auto‐evaluates for simple fractions of , for more complicated cases it stays as a cosine function to avoid the build up of large expressions. Using the function FunctionExpand, the cosine function can sometimes be transformed into explicit radicals. Here are some examples.

If the denominator contains squares of integers other than 2, the results always contain complex numbers (meaning that the imaginary number appears unavoidably).

The function RootReduce allows for writing the last algebraic numbers as roots of polynomial equations.

The function FunctionExpand also reduces trigonometric expressions with compound arguments or compositions, including inverse trigonometric functions, to simpler ones. Here are some examples.

Applying Simplify to the last expression gives a more compact result.

FullSimplify

The function FullSimplify tries a wider range of transformations than Simplify and returns the simplest form it finds. Here are some examples that contrast the results of applying the functions Simplify and FullSimplify to the same expressions.

Operations carried out by specialized Mathematica functions

Series expansions

Calculating the series expansion of a cosine function to hundreds of terms can be done in seconds.

Mathematica comes with the add‐on package DiscreteMath`RSolve` that allows finding the general terms of the series for many functions. After loading this package, and using the package function SeriesTerm, the following term of can be evaluated.

This result can be checked by the following process.

Differentiation

Mathematica can evaluate derivatives of the cosine function of an arbitrary positive integer order.

Finite summation

Mathematica can calculate finite symbolic sums that contain the cosine function. Here are some examples.

Infinite summation

Mathematica can calculate infinite sums including the cosine function. Here are some examples.

Finite products

Mathematica can calculate some finite symbolic products that contain the cosine function. Here is an example.

Indefinite integration

Mathematica can calculate a huge number of doable indefinite integrals that contain the cosine function. Here are some examples.

Definite integration

Mathematica can calculate wide classes of definite integrals that contain the cosine function. Here are some examples.

Limit operation

Mathematica can calculate limits that contain the cosine function. Here are some examples.

Solving equations

The next inputs solve two equations that contain the cosine function. Because of the multivalued nature of the inverse cosine function, a printed message indicates that only some of the possible solutions are returned.

A complete solution of the previous equation can be obtained using the function Reduce.

Solving differential equations

Here are differential equations whose linear independent solutions include the cosine function. The solutions of the simplest second-order linear ordinary differential equation with constant coefficients can be represented using and .

In the last input, the differential equation was solved for . If the argument is suppressed, the result is returned as a pure function (in the sense of the ‐calculus).

The advantage of such a pure function is that it can be used for different arguments, derivatives, and more.

In carrying out the algorithm to solve the following nonlinear differential equation, Mathematica has to solve a transcendental equation. In doing so, the generically multivariate inverse of a function is encountered, and a message is issued that a solution branch is potentially missed.

Integral transforms

Mathematica supports the main integral transforms like direct and inverse Fourier, Laplace, and Z transforms that can give results that contain classical or generalized functions.

Plotting

Mathematica has built‐in functions for 2D and 3D graphics. Here are some examples.