Tangent

The trigonometric function

$$y=\tan x=\frac{\sin x}{\cos x};$$

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:

$$\int\tan xdx=-\ln|\cos x|+c.$$

The tangent has a series expansion:

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

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