Tangent function

This article is about a particular function from a subset of the real numbers to the real numbers. Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article.
View a complete list of particular functions on this wiki

For functions involving angles (trigonometric functions, inverse trigonometric functions, etc.) we follow the convention that all angles are measured in radians. Thus, for instance, the angle of $90\,^{\circ }$ is measured as $\pi /2$.

Definition

Definition in terms of sine and cosine

The tangent function, denoted $\tan$, is defined as the quotient of the sine function by the cosine function, and it is defined wherever the cosine function takes a nonzero value. In symbols:

$\!\tan x:={\frac {\sin x}{\cos x}}$

Definition in terms of the unit circle

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Definition for acute angles in terms of triangles

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Key data

Item	Value
default domain	all odd integer multiples of $\pi /2$.
range	all real numbers, i.e., all of $\mathbb {R}$
period	$\pi$, i.e., $180\,^{\circ }$
local maximum values and points of attainment	there are no local maximum values
local minimum values and points of attainment	there are no local minimum values
points of inflection (both coordinates)	all points of the form $(n\pi ,0)$ where $n$ varies over integers.
vertical asymptotes	All lines of the form $x=n\pi +\pi /2$, $n$ varies over integers. At each such line, the left hand limit is $+\infty$ and the right hand limit is $-\infty$.
important symmetries	odd function half turn symmetry about all points of the form $(n\pi +\pi /2,0)$ where $n$ varies over integers.
first derivative	$x\mapsto \sec ^{2}x$, i.e., the secant-squared function. Note that $\sec ^{2}x=1+\tan ^{2}x$.
second derivative	$x\mapsto 2\sec ^{2}x\tan x$.
higher derivatives	every derivative can be expressed as a polynomial in terms of $\tan$. The degree of the $n^{th}$ derivative as a polynomial in $\tan$ is $n+1$.
first antiderivative	$-\ln |\cos x|+C=\ln |\sec x|+C$.
higher antiderivatives	no antiderivatives higher than the first are expressible in terms of elementary functions.

Integration

First antiderivative: using f'/f formulation

We use the integration form integration of quotient of derivative of function by function:

$\int {\frac {f'(x)\,dx}{f(x)}}=\ln |f(x)|+C$

In our case, we write:

$\int \tan x\,dx=\int {\frac {\sin x}{\cos x}}\,dx=-\int {\frac {-\sin x}{\cos x}}\,dx$

Using the integration form above with $f=\cos$, we get:

$-\ln |\cos x|+C$

Alternative method: [SHOW MORE]

Multiply and divide by $\sec x$ to get:

$\int \tan x\,dx=\int {\frac {\sec x\tan x}{\sec x}}\,dx$

We thus obtain a $f'/f$ form with $f=\sec$, and we get:

$\ln |\sec x|+C$

Repeated antidifferentiation

It is not possible to antidifferentiate the first antiderivative of $\tan$ within the universe of elementarily expressible functions.