Encyclopedia
In
mathematics, the word
tangent has two distinct but etymologically-related meanings: one in
geometry and one in
trigonometry.
Geometry
In plane geometry, a straight line is
tangent to a curve, at some point, if both line and curve pass through the point with the same direction; such a line is the best straight-line approximation to the curve at that point. The curve, at point
P, has the same slope as a tangent passing through
P. The slope of a
tangent line can be approximated by a
secant line. It is a mistake to think of tangents as lines which intersect a curve at only one single point. There are tangents which intersect curves at several points , and there are non-tangential lines which intersect curves at only one single point. It is also possible for a line to be a
double tangent, when it is tangent to the same curve at two distinct points. Higher numbers of tangent points are possible as well.
In the following diagram, a red line intersects the black curve at two points. It is tangent to the curve at the point indicated by the dot.
In higher-dimensional geometry, one can define the
tangent plane for a
surface in an analogous way to the tangent line for a curve. In general, one can have an -dimensional
tangent hyperplane to an
n-dimensional
manifold.
Quotation
"And I dare say that this is not only the most useful and general [concept] in geometry, that I know, but even that I ever desire to know."
DescartesCalculus
A "formal" definition of the tangent requires
calculus. Specifically, suppose a curve is the graph of some function,
y =
f, and we are interested in the point where
y0 =
f. The curve has a non-vertical tangent at the point if and only if the function is
differentiable at
x0. In this case, the
slope of the tangent is given by
f '. The curve has a vertical tangent at if and only if the slope approaches plus or minus
infinity as one approaches the point from either side.
Above, it was noted that a secant can be used to approximate a tangent; it could be said that the slope of a secant approaches the slope of the tangent, as the secants' points of intersection approach each other. Should one also understand the notion of a limit; one might understand how that concept is applicable to those discussed here, via
calculus. In essence, calculus was developed as a means to find the slopes of tangents; this challenge, being known as the
tangent line problem, is solvable via
Newton's
difference quotient.
Should one know the slope of a tangent, to some function; then, one can determine an equation for the tangent. For example, an understanding of the power rule will help one determine that the slope of
x3, at
x = 2, is 12. Using the
point-slope equation, one can write an equation for this tangent:
y − 8 = 12 = 12
x − 24; or: y = 12
x − 16.
Trigonometry
In
trigonometry, the
tangent is a function defined as:
The function is so-named because it can be defined as the length of a certain segment of a tangent to the
unit circle. It is easiest to define it in the context of a two-dimensional
Cartesian coordinate system. If one constructs the unit circle centered at the origin, the tangent line to the unit circle at the point P = , and the ray emanating from the origin at an angle ? to the
x-axis, then the ray will intersect the tangent line at
at most a single point Q. The tangent of ? is the length of the portion of the tangent line between P and Q. If the ray does not intersect the tangent line, then the tangent of ? is undefined.
Tangent was introduced by the
danish mathematician Thomas Fincke in his book Geometria rotundi .
The trigonometric tangent function arises as a generating function in combinatorics; see alternating permutation.
Derivative
The derivative of the tangent is :
Power series
See also the
list of Taylor series of some common functions.
See also
External links
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