Tangent -- From Wolfram MathWorld

The tangent function is defined by

(1)

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

The common schoolbook definition of the tangent of an angle in a right triangle (which is equivalent to the definition just given) is as the ratio of the side lengths opposite to the angle and adjacent the angle, i.e.,

(2)

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 has an important related meaning as a line or plane which touches a given curve or solid at a single point. These geometrical objects are then called a tangent line or tangent plane, respectively.

The definition of the tangent function can be extended to complex arguments using the definition

			(3)
			(4)
			(5)
			(6)

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

A related function known as the hyperbolic tangent is similarly defined,

(7)

An important tangent identity is given by

(8)

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

			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)

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

			(16)
			(17)

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

(18)

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

(19)

Another tangent identity is

			(20)
			(21)
			(22)

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

(23)

This gives the first few expansions as

			(24)
			(25)
			(26)
			(27)
			(28)

(OEIS A034867 and A034839).

A beautiful formula that generalizes the tangent angle addition formula, (27), and (28) is given by

(29)

(Szmulowicz 2005).

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

(30)

(Borchardt and Perrott 1930).

The Maclaurin series valid for for the tangent function is

			(31)
			(32)

(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

(33)

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

(34)

both due to Lambert.

An interesting identity involving the product of tangents is

(35)

where is the floor function.

The equation

(36)

which is equivalent to , where is the tanc function, does not have simple closed-form solutions.

The difference between consecutive solutions gets closer and closer to for higher order solutions. The function is sometimes known as the tanc function.