Tangent

The tangent function is defined for real numbers with by

where is the sine function and is the cosine function. The notation is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix).

For an acute angle in a right triangle, its tangent is the ratio of the side lengths opposite to the angle and adjacent to the angle, i.e.,

A convenient mnemonic for remembering the definition of the sine, cosine, and tangent is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent).

The word "tangent" also occurs in the geometric concepts of a tangent line to a curve and a tangent plane to a surface. Tangency does not require that the objects have only one point in common.

The definition of the tangent function extends to complex numbers with as

where e is the base of the natural logarithm and i is the imaginary unit. The tangent is implemented in the Wolfram Language as Tan[z].

A related function known as the hyperbolic tangent is defined for with by

The following identities hold wherever their displayed fractions are defined. An important tangent identity is

Angle addition formulas, subtraction and half-angle identities, and multiple-angle formulas are given by

The sine and cosine functions can conveniently be expressed in terms of a tangent as

which can be particularly convenient in polynomial computations such as Gröbner basis since it reduces the number of equations compared with explicit inclusion of and together with the additional relation (Trott 2006, p. 39).

These lead to the pretty identity

There is also a beautiful angle addition identity for three variables,

For a positive integer , another tangent identity is

where (Beeler et al. 1972). Written explicitly,

This gives the first few expansions as

(OEIS A034867 and A034839).

For a positive integer , the angle addition formulas generalize to

(Szmulowicz 2005).

There are a number of simple but interesting tangent identities based on those given above, including

(Borchardt and Perrott 1930).

The Maclaurin series for the tangent function, valid for , is

(OEIS A002430 and A036279), where is a Bernoulli number.

is irrational for any rational , which can be proved by writing as a continued fraction as

(Wall 1948, p. 349; Olds 1963, p. 138) and

both due to Lambert.

For a positive integer , an identity involving the product of tangents is

where is the floor function.

The equation

is equivalent to , where is the tanc function with . It has the solution and pairs of positive and negative real roots. The difference between consecutive positive roots tends to .