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Tangent

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The trigonometric function

another notation is: tg. Its domain of definition is the entire number axis with the exception of the points \pi/2=n\pi, n=\pm1,\pm2,\dots. The tangent is an unbounded, odd and periodic (with \pi as the smallest positive period) function. The tangent and the cotangent are connected by the relation

\tan x=\frac{1}{\operatorname{cotan}x}

The inverse function to the tangent is called the arctangent.

The derivative of the tangent is:

(\tan x)'=\frac{1}{\cos^2x}.

The indefinite integral of the tangent is:

\int\tan x\,dx=-\ln|{\cos x}|+c.

The tangent has a series expansion:

\tan x=x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{315}+\dotsb,\quad|x|<\frac\pi2.

The tangent of a complex argument z is a meromorphic function with zeros at the points z=k\pi, where k=0,\pm1,\pm2,\dots.


Comments

The general term in the series expansion of the tangent is:

\frac{2^{2n}(2^{2n}-1)|B_{2n}|}{(2n)!}x^{2n-1},

where B_{2n} are the Bernoulli numbers.

See also Trigonometric functions.

The addition formula of the tangent is:

\tan(x_1+x_2)=\frac{\tan x_1+\tan x_2}{1-\tan x_1\tan x_2}.

References

[a1] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1965) pp. 71ff
How to Cite This Entry:
Tangent. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent&oldid=44649
This article was adapted from an original article by Yu.A. Gor'kov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article