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.
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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

Fill this in later

Definition for acute angles in terms of triangles

Suppose $t$ is an acute angle, i.e., $t\in (0,\pi /2)$. Construct a right triangle where $t$ is one of the acute angles. $\tan t$ is defined as the quotient of the leg opposite $t$ to the leg adjacent to $t$.

Key data

Item	Value
default domain	all real numbers except odd integer multiples of $\pi /2$, i.e., $\mathbb{R}\setminus \{n\pi +\pi /2\mid n\in \mathbb{Z}\}$. This is a union of countably many open intervals of the form $(n\pi -\pi /2,n\pi +\pi /2)$ with $n\in \mathbb{Z}$: $\dots \cup (-3\pi /2,-\pi /2)\cup (-\pi /2,\pi /2)\cup (\pi /2,3\pi /2)\cup \dots$
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 $+\mathrm{\infty }$ and the right hand limit is $-\mathrm{\infty }$.
important symmetries	odd function half turn symmetry about all points of the form $(n\pi /2,0)$ where $n$ varies over integers. Note that for the even multiples of $\pi /2$, (i.e., the multiples of $\pi$, this is half turn symmetry about points of inflection. For the odd multiples of $\pi /2$, this is half turn symmetry about points not on the graph itself.
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	$-\ln \left|\cos x\right|+C=\ln \left|\sec x\right|+C$.
higher antiderivatives	no antiderivatives higher than the first are expressible in terms of elementary functions.

Differentiation

First derivative

WHAT WE USE: sine function#First derivative, cosine function#First derivative, quotient rule for differentiation

The first derivative is:

$\frac{d}{dx}(\tan x)={\sec }^{2}x$

Here's how we get it. We start with:

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

Using the quotient rule for differentiation, we get:

${{\tan }^{\prime }}^{\prime }x=\frac{(\cos x)({{\sin }^{\prime }}^{\prime }x)-(\sin x)({{\cos }^{\prime }}^{\prime }x)}{{\cos }^{2}x}=\frac{(\cos x)(\cos x)-(\sin x)(-\sin x)}{{\cos }^{2}x}=\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}={\sec }^{2}x$

The penultimate step uses ${\cos }^{2}x+{\sin }^{2}x=1$.

Second derivative

WHAT WE USE: secant function#First derivative, chain rule for differentiation, differentiation rule for power functions

The second derivative is:

$\frac{d^2}{d{x}^2}(\tan x)=2\tan x{\sec }^{2}x$

We obtain this by differentiating the first derivative:

Failed to parse (syntax error): {\displaystyle \frac{{d^2}{dx^2}(\tan x) = \frac{d}{dx}(\sec^2x) = \frac{d}{d(\sec x)}{(\sec^2x)} \frac{d(\sec x)}{dx} = 2\sec x (\sec x \tan x) = 2\sec^2x\tan x}

Integration

First antiderivative

{{WHAT WE USE: integration of quotient of derivative of function by function}}

We use the following form:

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

In our case, we write:

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

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

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

Alternative method: [SHOW MORE]

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

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

We thus obtain a $f^{\prime }/f$ form with $f=\sec$, and we get:

$\ln \left|\sec x\right|+C$

Definite integrals

The definite integral of the $\tan$ function can be computed on any closed interval that lies completely within one of the open intervals on which $\tan$ is defined, i.e., both endpoints must lie between the same pair of consecutive odd multiples of $\pi /2$. Thus, for instance, $\tan$ can be integrated from $\pi /4$ to $\pi /3$ but not from $\pi /3$ to $2\pi /3$, because these lie on opposite sides of the point $\pi /2$ at which the function has a vertical asymptote.

Moreover, all improper integrals are undefined.

If $a,b$ both lie between consecutive odd multiples of $\pi /2$, we get:

$\underset{a}{\overset{b}{\int }}\tan xdx=\ln (\frac{\cos a}{\cos b})$

Note that the lower limit comes on top because of the minus sign on the antiderivative. Further, we do not need to put an absolute value sign because $\cos$ has constant sign on each interval of definition, so the quotient is positive in sign.

Finally, note that $\tan$ is an odd function, and more generally, has half turn symmetry about points $(n\pi ,0)$, so:

$\underset{n\pi -t}{\overset{n\pi +t}{\int }}\tan xdx=0$

for $t\in (0,\pi /2)$.

Higher antiderivatives

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