tangent

In a conversation or creative process, a tangent is a line of reasoning followed off of the initial line. This whole informational archive is one giant quivering heap of tangents. Everytime you click on one of these hyperlinks, you're following a tangent.

a tangent is this:
nothing but a kiss.
a circle and a line
at a cosine.

In geometry, a tangent line is one that intersects a circle at exactly one point, just "grazing the edge" of it. (It comes from the Latin word tangere, "to touch.") Compare to secant.

In trigonometry, the tangent (tan) function is so named because of this geometric representation. If you draw a unit circle on a pair of coordinate axes and then add a line segment out from its center, that line will strike the unit circle at one point. The y-coordinate of that point is the sine of the angle formed by the line segment, and its x-coordinate is the cosine of the angle.

                 cosine
                   _/\_
                 |/    \
            _____|_____     _
           /     |   /|\     \
          /      |  / | \     } sine
         |       | /  |  |  _/
_________|_______|/___|__|_________
         |       |       |
         |       |       |
          \      |      /
           \_____|_____/
                 |
                 |

If you extend this line segment beyond the unit circle, far enough that you can drop a vertical line that is tangent to the circle, you create a new triangle which is similar to the one inside the unit circle:

                       
                        /|
                 |     / |
            _____|____/  |   
           /     |   /|\ |    
          /      |  / | \|          
         |       | /  |  |    
_________|_______|/___|__|_________
         |       |       |
         |       |       |
          \      |      /
           \_____|_____/
                 |
                 |

Because the triangles are similar, the ratios of their sides are equal. Since we are dealing with a unit circle, its radius is 1. Therefore the length of the tangent segment (divided by 1) equals the ratio of the sine to the cosine.

Proof that a tangent is perpendicular to the radius drawn to the point of tangency:

Givens are circle O, line AB tangent to O at point A, and radius OA.

We'll use an indirect proof; that is, we'll begin by assuming that OA is not perpendicular to AB. Therefore, there must be some line segment that is perpendicular to AB; we'll call it OX (where X is a point on AB). A perpendicular line is the shortest distance to a point from a line; therefore, OX is the shortest distance to O from AB, and is shorter than OA. But a radius is always shorter than a line segment from the center of a circle to a point in the exterior of the circle, so OA is shorter than OX. We have reached a contradiction, and have proved the initial assumption false. Q.E.D.

This is a graph of the equation y = tan x. Dotted lines indicate discontinuities.

+-------------------------------------------------------------------------------------------------+
|                   #    .                       |                  #    .                        |  2.5
|                  #     .                       |                 #     .                        |
|                 ##     .                       |                ##     .                        |
|                 #      .                       |                #      .                        |  2.0
|                ##      .                       |               ##      .                        |
|                #       .                       |               #       .                        |
|               #        .                       |              #        .                        |  1.5
|              #         .                       |             #         .                        |
|             #          .                       |            #          .                        |
|            #           .                       |           #           .                        |  1.0
|          ##            .                       |         ##            .                        |
|         #              .                       |        #              .                        |
|       ##               .                       |      ##               .                        |  0.5
|    ###                 .                       |   ###                 .                        |
|  ##                    .                       | ##                    .                        |
|##---------------------------------------------###---------------------------------------------##|  0.0
|                        .                    ## |                       .                    ##  |
|                        .                 ###   |                       .                 ###    |
|                        .               ##      |                       .               ##       | -0.5
|                        .              #        |                       .              #         |
|                        .            ##         |                       .            ##          |
|                        .           #           |                       .           #            | -1.0
|                        .          #            |                       .          #             |
|                        .         #             |                       .         #              |
|                        .        #              |                       .        #               | -1.5
|                        .       #               |                       .       #                |
|                        .      ##               |                       .      ##                |
|                        .      #                |                       .      #                 | -2.0
|                        .     ##                |                       .     ##                 |
|                        .     #                 |                       .     #                  |
|                        .    #                  |                       .    #                   | -2.5
+-------------------------------------------------------------------------------------------------+
 A                       A                       A                       A                       A
-π                     -π/2                      0                      π/2                      π

Tan"gent (?), n. [L. tangens, -entis, p.pr. of tangere to touch; akin to Gr. having seized: cf. F. tangente. Cf. Attain, Contaminate, Contingent, Entire, Tact, Taste, Tax, v. t.] Geom.

A tangent line curve, or surface; specifically, that portion of the straight line tangent to a curve that is between the point of tangency and a given line, the given line being, for example, the axis of abscissas, or a radius of a circle produced. See Trigonometrical function, under Function.

Artificial, or Logarithmic, tangent, the logarithm of the natural tangent of an arc. -- Natural tangent, a decimal expressing the length of the tangent of an arc, the radius being reckoned unity. -- Tangent galvanometer Elec., a form of galvanometer having a circular coil and a short needle, in which the tangent of the angle of deflection of the needle is proportional to the strength of the current. -- Tangent of an angle, the natural tangent of the arc subtending or measuring the angle. -- Tangent of an arc, a right line, as ta, touching the arc of a circle at one extremity a, and terminated by a line ct, passing from the center through the other extremity o. <-- references are to a figure showing the tangent of an arc -->

 

© Webster 1913.

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Tan"gent, a. [L. tangens, -entis, p.pr.]

Touching; touching at a single point

; specifically Geom.

meeting a curve or surface at a point and having at that point the same direction as the curve or surface; -- said of a straight line, curve, or surface; as, a line tangent to a curve; a curve tangent to a surface; tangent surfaces.

Tangent plane Geom., a plane which touches a surface in a point or line. -- Tangent scale Gun., a kind of breech sight for a cannon. -- Tangent screw Mach., an endless screw; a worm.

 

© Webster 1913.