Introduction to the Tangent Function

Defining the tangent function

The tangent function is an old mathematical function. It was mentioned in 1583 by T. Fincke who introduced the word "tangens" in Latin. E. Gunter (1624) used the notation "tan", and J. H. Lambert (1770) discovered the continued fraction representation of this function.

The classical definition of the tangent function for real arguments is: "the tangent of an angle in a right‐angle triangle is the ratio of the length of the opposite leg to the length of the adjacent leg." This description of is valid for when the triangle is nondegenerate. This approach to the tangent can be expanded to arbitrary real values of if consideration is given to the arbitrary point in the ,‐Cartesian plane and is defined as the ratio , assuming that α is the value of the angle between the positive direction of the ‐axis and the direction from the origin to the point .

Comparing this definition with definitions of the sine and cosine functions shows that the following formula can also be used as a definition of the tangent function:

A quick look at the tangent function

Here is a graphic of the tangent function for real values of its argument .

Representation through more general functions

The tangent function can be represented using more general mathematical functions. As the ratio of the sine and cosine functions that are particular cases of the generalized hypergeometric, Bessel, Struve, and Mathieu functions, the tangent function can also be represented as ratios of those special functions. But these representations are not very useful. It is more useful to write the tangent function as particular cases of one special function. That can be done using doubly periodic Jacobi elliptic functions that degenerate into the tangent function when their second parameter is equal to or :

Definition of the tangent function for a complex argument

In the complex ‐plane, the function is defined using and or the exponential function in the points and through the formula:

In the points , where has zeros, the denominator of the last formula equals zero and has singularities (poles of the first order).

Here are two graphics showing the real and imaginary parts of the tangent function over the complex plane.

The best-known properties and formulas for the tangent function

Values in points

Students usually learn the following basic table of tangent function values for special points of the circle:

General characteristics

For real values of argument , the values of are real.

In the points , the values of are algebraic. In several cases they can be integers , 0, or 1:

The values of can be expressed using only square roots if and is a product of a power of 2 and distinct Fermat primes {3, 5, 17, 257, …}.

The function is an analytical function of that is defined over the whole complex ‐plane and does not have branch cuts and branch points. It has an infinite set of singular points:

(a) are the simple poles with residues –1. (b) is an essential singular point.

It is a periodic function with the real period :

The function is an odd function with mirror symmetry:

Differentiation

The first derivative of has simple representations using either the function or the function:

The derivative of has much more complicated representations than symbolic derivatives for and :

where is the Kronecker delta symbol: and .

Ordinary differential equation

The function satisfies the following first-order nonlinear differential equation:

Series representation

The function has a simple series expansion at the origin that converges for all finite values with :

where are the Bernoulli numbers.

Integral representation

The function has a well-known integral representation through the following definite integral along the positive part of the real axis:

Continued fraction representations

The function has the following simple continued fraction representations:

Indefinite integration

Indefinite integrals of expressions involving the tangent function can sometimes be expressed using elementary functions. However, special functions are frequently needed to express the results even when the integrands have a simple form (if they can be evaluated in closed form). Here are some examples:

Definite integration

Definite integrals that contain the tangent function are sometimes simple. For example, the famous Catalan constant can be defined as the value of the following integral:

This constant also appears in the following integral:

Some special functions can be used to evaluate more complicated definite integrals. For example, the generalized hypergeometric and polygamma functions are needed to express the following integral:

Finite summation

The following finite sums that contain the tangent function can be expressed using cotangent functions:

Other finite sums that contain the tangent function can be expressed using polynomial functions:

Infinite summation

The evaluation limit of the first formula from the previous subsubsection for gives the following value for the corresponding infinite sum from the tangent:

Other infinite sums that contain the tangent can also be expressed using elementary functions:

Finite products

The following finite product from the tangent has a very simple value:

Addition formulas

The tangent of a sum can be represented by the rule: "the tangent of a sum is equal to the sum of tangents divided by one minus the product of tangents." A similar rule is valid for the tangent of the difference:

Multiple arguments

In the case of multiple arguments , , , …, the function can be represented as the ratio of the finite sums including powers of tangents:

Half-angle formulas

The tangent of the half‐angle can be represented using two trigonometric functions by the following simple formulas:

The sine function in the last formula can be replaced by the cosine function. But it leads to a more complicated representation that is valid in some vertical strips:

To make this formula correct for all complex , a complicated prefactor is needed:

where contains the unit step, real part, imaginary part, the floor, and the round functions.

Sums of two direct functions

The sum of two tangent functions can be described by the rule: "the sum of tangents is equal to the sine of the sum multiplied by the secants." A similar rule is valid for the difference of two tangents:

Products involving the direct function

The product of two tangent functions and the product of the tangent and cotangent have the following representations:

Inequalities

The most famous inequality for the tangent function is the following:

Relations with its inverse function

There are simple relations between the function and its inverse function :

The second formula is valid at least in the vertical strip . Outside of this strip a much more complicated relation (that contain the unit step, real part, and the floor functions) holds:

Representations through other trigonometric functions

Tangent and cotangent functions are connected by a very simple formula that contains the linear function in the argument:

The tangent function can also be represented using other trigonometric functions by the following formulas:

Representations through hyperbolic functions

The tangent function has representations using the hyperbolic functions:

Applications

The tangent function is used throughout mathematics, the exact sciences, and engineering.