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 is measured as .
- 1 Definition
- 2 Key data
- 3 Identities
- 4 Values
- 5 Differentiation
- 6 Points and intervals of interest
- 7 Integration
- 8 Differential equations
- 9 Power series and Taylor series
- 10 Limit computations
Definition in terms of sine and cosine
Unit circle definition
The tangent function, denoted , is defined as follows.
Consider the unit circle centered at the origin, described as the following subset of the coordinate:
For a real number , we define as follows:
- Start at the point , which lies on the unit circle centered at the origin.
- Move a distance of along the unit circle in the counter-clockwise direction (i.e., the motion begins in the first quadrant, with both coordinates positive).
- At the end, the quotient of the -coordinate by the -coordinate of the point thus obtained is defined as .
Definition for acute angles in terms of triangles
Suppose is an acute angle, i.e., . Construct a right triangle where is one of the acute angles. is defined as the quotient of the leg opposite to the leg adjacent to .
|default domain|| all real numbers except odd integer multiples of , i.e., .|
This is a union of countably many open intervals of the form with :
|range||all real numbers, i.e., all of|
|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 where varies over integers.|
|vertical asymptotes||All lines of the form , varies over integers. At each such line, the left hand limit is and the right hand limit is .|
|important symmetries|| odd function|
half turn symmetry about all points of the form where varies over integers. Note that for the even multiples of , (i.e., the multiples of , this is half turn symmetry about points of inflection. For the odd multiples of , this is half turn symmetry about points not on the graph itself.
|first derivative||, i.e., the secant-squared function. Note that .|
|higher derivatives||every derivative can be expressed as a polynomial in terms of . The degree of the derivative as a polynomial in is .|
|higher antiderivatives||no antiderivatives higher than the first are expressible in terms of elementary functions.|
|Type of identity||Identity in algebraic form|
|complementary angle||, relates to cotangent function, its composite with reciprocal function|
|square relationship with secant|
|angle sum tangent formula|
|angle difference tangent formula|
|double angle tangent formula|
|formula for double angle sine and cosine in terms of tangent|
|formula for tangent of acute angle in terms of double angle cosine|
|other symmetries|| periodicity: |
Values at some acute angles
All these can be deduced from the values at and and the identities above. The values at and in turn can be deduced from the corresponding sine or cosine values.
|Angle in radians (default)||Numerical approximation for angle in radians||Angle in degrees||Formal expression for||Numerical approximation|
Note that for close to zero, is just slightly bigger than . As becomes bigger, the functions diverge and as , .
WHAT WE USE: sine function#First derivative, cosine function#First derivative, quotient rule for differentiation
The first derivative is:
Here's how we get it. We start with:
Using the quotient rule for differentiation, we get:
The penultimate step uses .
WHAT WE USE: secant function#First derivative, chain rule for differentiation, differentiation rule for power functions
The second derivative is:
We obtain this by differentiating the first derivative:
Points and intervals of interest
Consider the function .
At each of the points where is undefined, which are precisely the odd multiples of , the left hand limit is and the right hand limit is .
As computed earlier, we have:
The original function is undefined at odd multiples of . The expression for is also undefined at precisely these points, but these are not considered critical points. Note that is defined wherever is. Thus, the only kind of critical points are those where . However, anywhere because forces .
The upshot is that the function has no critical points on its domain of definition.
Intervals of increase and decrease
Wherever the function is defined, the derivative is positive. Thus, is increasing on each of the open intervals in its domain of definition, i.e., is increasing on each of the intervals:
Further, on each interval, it increases from a limiting value of at the left end of the interval to a limiting value of at the right end of the interval.
However, it is not correct to say that is increasing throughout its domain. This is because between successive intervals, it jumps from <math+\infty</math> to .
Local extreme values
There are no local extreme values for the function.
Intervals of concave up and concave down
The second derivative is . The sign of this is determined by the sign of . Thus, when , the graph is concave down, and when , the graph is concave up. Unpacking, we get:
- The graph of is concave down on intervals of the form .
- The graph of is concave up on intervals of the form .
Points of inflection
The points of inflection on the graph are points of the form , i.e., integer multiples of . At these points, the graph transitions from concave down (on the left) to concave up (on the right).
We use the following form:
In our case, we write:
Using the integration form above with , we get:
Alternative method: [SHOW MORE]
The definite integral of the function can be computed on any closed interval that lies completely within one of the open intervals on which is defined, i.e., both endpoints must lie between the same pair of consecutive odd multiples of . Thus, for instance, can be integrated from to but not from to , because these lie on opposite sides of the point at which the function has a vertical asymptote.
Moreover, all improper integrals are undefined.
If both lie between consecutive odd multiples of , we get:
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 has constant sign on each interval of definition, so the quotient is positive in sign.
Finally, note that is an odd function, and more generally, has half turn symmetry about points , so:
We can use the integration of to integrate any function of the form using the integration of linear transform of function:
The points where the transformed function is undefined are odd multiples of minus , i.e.,:
It is not possible to antidifferentiate the first antiderivative of within the universe of elementarily expressible functions.
Autonomous differential equations with this function as solution
The standard differential equation is:
The general solution to this is where is a constant. The value of can be determined using initial value conditions.
Power series and Taylor series
Computation of Taylor series
The function has a Taylor series at . The first few terms are indicated below:
Note the following:
- All the terms with nonzero coefficients are odd degree terms, because is an odd function.
- All the coefficients on odd degree terms are positive. This follows from the fact that all the higher derivatives of , evaluated at 0, are positive.
The coefficients can be written explicitly using the odd-degree up/down numbers, which are also called the tangent numbers or zag numbers. Explicitly, if denotes the tangent number, then the Taylor series for is:
Taylor series equals power series
The Taylor series for converges to on its interval of convergence, which is .
Order of zero
We get the following limit from the power series:
Thus, the order of the zero of at 0 is 1 and the residue is 1.
The limit can be computed in many ways:
|Name of method for computing the limit||Details|
|Simple manipulation, using|
|Using the L'Hopital rule|
|Using the power series||We have , so . Taking the limit at gives 1.|