# Dictionary Definition

tangent

### Noun

1 a straight line or plane that touches a curve
or curved surface at a point but does not intersect it at that
point

2 ratio of the opposite to the adjacent side of a
right-angled triangle [syn: tan]

# User Contributed Dictionary

## English

### Pronunciation

- /'tæn.dʒənt/

### Etymology

From tangentem, from tangere.### Noun

- A topic nearly
unrelated to the main
topic, but having a point in common with it.
- I believe we went off onto a tangent when we started talking about unicycles.

- A small metal blade by which a clavichord produces sound.

#### Derived terms

#### Related terms

#### Translations

in geometry

in trigonometry

nearly unrelated topic

- Catalan: tangent
- Finnish: sivuraide, harhapolku
- Polish: punkt wspólny

### Adjective

- Touching a curve at a single point but not crossing it at that point.
- Of a topic, only loosely related to a main topic.

### See also

# Extensive Definition

- For the tangent function see trigonometric functions. For other uses, see tangent (disambiguation).

In a similar way, the tangent plane to a surface at a given point is the
plane
that "just touches" the surface at that point. The concept of a
tangent is one of the most fundamental notions in differential
geometry and has been extensively generalized — see Tangent
space.

The word "tangent" comes from the Latin tangere,
meaning "to touch".

## Tangent line to a curve

The intuitive notion that a tangent "just
touches" a curve can be made more explicit by considering the
sequence of straight lines (secant lines)
passing through two points, A and B, that lie on the curve. The
tangent at A is the limit of the progression of secant lines as B
moves ever closer to A. The existence and uniqueness of the tangent
line depends on a certain type of mathematical smoothness, known as
"differentiability". For example, if two circular arcs meet at a
sharp point (a vertex) then there is no uniquely defined tangent at
the vertex because the limit of the progression of secant lines
depends on the direction in which "point B" approaches the
vertex.

In most cases commonly encountered, the tangent
to a curve does not cross the curve at the point of tangency
(though it may, when continued, cross the curve at other places
away from the point of tangency). This is true, for example, of all
tangents to a circle or a
parabola. However, at
exceptional points called inflection
points, the tangent line does cross the curve at the point of
tangency. An example is the point (0,0) on the graph of the cubic
parabola y = x3.

Conversely, it may happen that the curve lies
entirely on one side of a straight line passing through a point on
it, and yet this straight line is not a tangent line. This is the
case, for example, for a line passing through the vertex of a
triangle and not
interecting the triangle — where the tangent line does
not exist for the reasons explained above. In convex
geometry, such lines are called supporting
lines.

### Analytical approach

The geometric idea of the tangent line as the
limit of secant lines serves as the motivation for analytical
methods that are used to find tangent lines explicitly. The
question of finding the tangent line to a graph, or the tangent
line problem, was one of the central questions leading to the
development of calculus
in the 17th century. In the second book of his Geometry,
René
Descartes
said of the problem of constructing the tangent to a curve,
"And I dare say that this is not only the most useful and most
general problem in geometry that I know, but even that I have ever
desired to know"

#### Intuitive description

Suppose that a curve is given as the graph of a
function,
y = f(x). To find the tangent line at the point p = (a, f(a)),
consider another nearby point q = (a + h, f(a + h)) on the curve.
The slope of the secant line
passing through p and q is equal to the difference
quotient

- \frac.

As the point q approaches p, which corresponds to
making h smaller and smaller, the difference quotient should
approach a certain limiting value k, which is the slope of the
tangent line at the point p. If k is known, the equation of the
tangent line can be found in the point-slope form:

- y-f(a) = k(x-a).\,

#### More rigorous description

To make the preceding reasoning rigorous, one has
to explain what is meant by the difference quotient approaching a
certain limiting value k. The precise mathematical formulation was
given by Cauchy in the 19th
century and is based on the notion of limit.
Suppose that the graph does not have a break or a sharp edge at p
and it is neither plumb nor too wiggly near p. Then there is a
unique value of k such that as h approaches 0, the difference
quotient gets closer and closer to k, and the distance between them
becomes negligible compared with the size of h, if h is small
enough. This leads to the definition of the slope of the tangent
line to the graph as the limit of the difference quotients for the
function f. This limit is the
derivative of the function f at x = a, denoted
f ′(a). Using derivatives, the equation of the
tangent line can be stated as follows:

- y=f(a)+f'(a)(x-a).\,

Calculus provides rules for computing the
derivatives of functions that are given by formulas, such as the
power
function, trigonometric
functions, exponential
function, logarithm, and their various
combinations. Thus equations of the tangents to graphs of all these
functions, as well as many others, can be found by the methods of
calculus.

#### When the method fails

Calculus also demonstrates that there are
functions and points on their graphs for which the limit
determining the slope of the tangent line does not exist. For these
points the function f is non-differentiable. There are two possible
reasons for the method of finding the tangents based on the limits
and derivatives to fail: either the geometric tangent exists, but
it is a vertical line, which cannot be given in the point-slope
form since it does not have a slope, or the graph is too badly
behaved to admit a geometric tangent.

The graph y = x1/3 illustrates the first
possibility: here the difference quotient at a = 0 is equal to
h1/3/h = h− 2/3, which becomes very large as h approaches
0. The tangent line to this curve at the origin is vertical.

The graph y = |x| of the absolute
value function consists of two straight lines with different
slopes joined at the origin. As a point q approaches the origin
from the right, the secant line always has slope 1. As a point q
approaches the origin from the left, the secant line always has
slope −1. Therefore, there is no unique tangent to the
graph at the origin (although in a certain sense, there are two
half-tangents, corresponding to two possible directions of
approaching the origin).

## Tangent circles

Two circles, with radii of ri and centers at (xi ,
yi), for i=1, 2 are said to be tangent to each other if

- \left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2=\left(r_1\pm r_2\right)^2

## Surfaces and higher-dimensional manifolds

The tangent plane to a surface at a given point p is
defined in an analogous way to the tangent line in the case of
curves. It is the best approximation of the surface by a plane at
p, and can be obtained as the limiting position of the planes
passing through 3 distinct points on the surface close to p as
these points converge to p. More generally, there is a
k-dimensional tangent
space at each point of a k-dimensional manifold in the n-dimensional
Euclidean
space.

## References

## See also

## External links

- MathWorld: Tangent Line
- Tangent to a circle With interactive animation
- Tangent and first derivative - An interactive simulation
- The Tangent Parabola by John H. Mathews

tangent in Arabic: ظل (رياضيات)

tangent in Bosnian: Tangens

tangent in Catalan: Tangent

tangent in Czech: Tečna

tangent in Danish: Tangent (Geometri)

tangent in German: Tangente

tangent in Spanish: Tangente

tangent in French: Tangente

tangent in Hebrew: משיק

tangent in Hungarian: Érintő

tangent in Indonesian: Tangen

tangent in Italian: Tangente

tangent in Dutch: Raaklijn

tangent in Norwegian: Tangent (matematikk)

tangent in Polish: Styczna

tangent in Portuguese: Tangente

tangent in Russian: Уравнение касательной

tangent in Slovak: Tangens

tangent in Swedish: Tangent

tangent in Chinese: 切线

# Synonyms, Antonyms and Related Words

abutter, adjoiner, air line, approach, asymptote, axis, beeline, borderer, bottleneck, bystander, chord, collision course, concentralization,
concentration,
concourse, concurrence, confluence, conflux, congress, convergence, converging, crossing, diagonal, diameter, direct line, directrix, edge, focalization, focus, funnel, great-circle course,
hub, immediate neighbor,
looker-on, meeting,
mutual approach, narrowing gap, neighbor, neighborer, normal, onlooker, perpendicular, radius, radius vector, right
line, secant, segment, shortcut, side, spokes, straight, straight course,
straight line, straight stretch, straightaway, streamline, transversal, vector