The tangent function is defined by

(1)

where is thesine function and is thecosine function. The notation is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix).

The common schoolbook definition of the tangent of an angle in aright 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 thesine, cosine, and tangent isSOHCAHTOA (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 aline orplane which touches a given curve or solid at a single point. These geometrical objects are then called atangent line ortangent plane, respectively.

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

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

wheree is the base of thenatural logarithm andi is theimaginary number. The tangent is implemented in theWolfram Language asTan[z].

A related function known as thehyperbolic tangentis 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)

Thesine andcosine functionscan conveniently be expressed in terms of a tangent as

			(16)
			(17)

which can be particularly convenient in polynomial computations such asGröbner 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 (Beeleret al.1972). Written explicitly,

(23)

This gives the first few expansions as

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

(OEISA034867 andA034839).

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).

TheMaclaurin series valid for for the tangent function is

			(31)
			(32)

(OEISA002430 andA036279), where is aBernoulli number.

isirrational for anyrational, which can be proved by writing as acontinued fraction as

(33)

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

(34)

both due to Lambert.

An interesting identity involving theproduct of tangentsis

(35)

where is thefloor function.

The equation

(36)

which is equivalent to, where is thetanc 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 thetanc function.

See also

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More things to try:

• tangent
• find the tangent line of y = (e^x)/x at (1, e)
• what is the equation of the tangent line to the curve x^2 + y^2 = 169 at (5, -12)

References

Szmulowicz, F. "New Analytic and Computational Formalism for the Band Structure of-Layer Photonic Crystals."Phys. Lett. A345, 469-477, 2005.

Spanier, J. and Oldham, K. B. "The Tangent and Cotangent Functions." Ch. 34 inAn Atlas of Functions. Washington, DC: Hemisphere, pp. 319-330, 1987.

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Tangent." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/Tangent.html

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