Define what a tangent line is.?

Question

Think about it a bit before you try to answer please.

Well it took 5 answers to get it lol... TY #5

Answer 1

It is the first order (linear) approximation to a curve at a given point. The definition "just touching but not croossing" makes sense for most of the points but it is not true in general. Say you have a sharp vertex, like the point at the origin for the graph of y=|x|. Then there are infinitely many lines that touch but not cross the graph at that point but no real tangent lines. Also the tangent to a straight line is itself which does not quite fit the touch and go approach above.

Answer 2

A tangent line is a straight line which intersects with, but does not cross a curved line or a curved surface. There may be more than one point of tangency, e.g. the x axis is tangent to versine x.

Correction:

Consider y=x^3, y=arctan x, and y=sin x. For each of these, the tangent at x=0 crosses the curve at the point of intersection. lines tangent at other points will also cross the curve at some point. Therefore the above definition is invalid and I have to resort to the slope definition:

A tangent line is a straight line whose slope is equal to the slope of a curve at the point of intersection. This can also be applied to curved surfaces.

Answer 3

In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry.

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 (as in the following example), and there are non-tangential lines which intersect curves at only one single point. (Note that in the important case of a conic section, such as a circle, the tangent line will intersect the curve at only one 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 (n − 1)-dimensional tangent hyperplane to an n-dimensional manifold.

A "formal" definition of the tangent requires calculus. Specifically, suppose a curve is the graph of some function, y = f(x), and we are interested in the point (x0, y0) where y0 = f(x0). The curve has a non-vertical tangent at the point (x0, y0) if and only if the function is differentiable at x0. In this case, the slope of the tangent is given by f '(x0). The curve has a vertical tangent at (x0, y0) 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 (or direction) 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 (in part) 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(x − 2) = 12x − 24; or: y = 12x − 16.

In trigonometry, the tangent is a function defined as:
tan x = sin x / coh x

The function is so-named because it can be defined as the length of a certain segment of a tangent (in the geometric sense) 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 = (1, 0), and the ray emanating from the origin at an angle θ to the x-axis, then the ray will intersect the tangent line at at most a single point Q. The tangent (in the trigonometric sense) of θ 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 (function) of θ is undefined.

Tangent was introduced by the danish mathematician Thomas Fincke in his book Geometria rotundi (1583).

The trigonometric tangent function arises as a generating function in combinatorics.

Answer 4

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 (as in the following example), and there are non-tangential lines which intersect curves at only one single point. (Note that in the important case of a conic section, such as a circle, the tangent line will intersect the curve at only one 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 (n − 1)-dimensional tangent hyperplane to an n-dimensional manifold.

Answer 5

Tangent (trigonometry), one of the six fundamental ratios of trigonometry. The others are sine, cosine, secant, cosecant, and cotangent. A ratio is a proportional relationship between two numbers calculated by dividing one number by the other. Tangent embodies such a relationship between the magnitudes of the angles of a right triangle (a triangle having one 90° angle) and the lengths of the sides. Varying one value, such as the magnitude of an angle, requires the related value, such as the length of a side, to change in a predictable way.
The tangent, usually abbreviated tan, of one of the acute (less than 90°) angles of a right triangle is found by dividing the length of the side opposite the angle by the length of the shorter of the two sides adjacent to the angle. (The other adjacent side is called the hypotenuse, and is the triangle’s longest side.)
If the acute angle in question is called theta (θ), then . Tangent smoothly increases in numerical value from 0 to infinity as the angle increases from 0° to 90°.
Tangent is also defined for angles greater than 90° using right triangles inscribed in a circle centered at the point (0,0) on the xy axis:
A line drawn from the circle’s center to any point on the circle makes an angle, θ, with the x axis. The tangent of θ is equal to the vertical distance of the point from the x axis divided by the horizontal distance of the point from the y axis. At 90°, tangent is discontinuous, flipping from positive infinity to negative infinity. The function rises from negative infinity to 0 at 180° and continues to climb, approaching positive infinity at 270°. At 270° tangent is again discontinuous, flipping from positive to negative infinity. Beyond 270° tangent increases in numerical value, reaching 0 at 360°.
Cotangent is tangent’s reciprocal function. The cotangent, usually abbreviated cot, of an acute angle of a right triangle is equal to the length of the shorter side adjacent to the chosen acute angle divided by the length of the side opposite the angle: .

Answer 6

when you say a line is tangent to a curve in point a for example it means the intersection of the line and the curve in that area is just in a point

Answer 7

It's a line that shares a common point on a curve and is the closest linear approximation to the surface (or 'top') of said curve.

Answer 8

a line that touches a circle at just one point is a tangent to a circle.It lies outside the circle

Answer 9

a line that touches the surface of a curve

Answer 10

http://en.wikipedia.org/wiki/Tangent_line (look at the "Geometry" section).