# Subtangent

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Subtangent

In geometry, the subtangent is the projection of the tangent upon the axis of abscissas (i.e., the "x"-axis).

"Tangent" here specifically means a line segment which is tangential to a point "P" on a curve and which intersects the "x"-axis at point "Q". The line segment "PQ" is the tangent, and the length of "PQ" is also called the "tangent".

Draw a line through "P" parallel to the axis of ordinates (a.k.a. "y"-axis). This line intersects the "x"-axis at "P' ". Then line "P'Q" is the "subtangent", and its length is also called the subtangent.

Let "&theta;" be the angle of inclination of the tangent with respect to the "x"-axis. Let the curve be described by "y=f(x)", let "x0" be the abscissa of point "P", and let "&theta;0" be the angle of inclination of the tangent of "P". Then this tangent of "P" is:$t = f\left(x_0\right) , csc heta_0 quad$and the subtangent is:$t_s = t , cos heta_0 = f\left(x_0\right) cot heta_0 quad$The angle of inclination "&theta;" is related to the derivative by:$heta = arctan \left\{df over dx\right\}$therefore:$t_s = \left\{ f\left(x_0\right) over f\text{'}\left(x_0\right) \right\}.$

The subtangent in polar coordinates

In polar coordinates, the tangent to a curve can be specifically defined as a line segment, tangential to the curve, which extends from the given point "P" on the curve to a point "T", such that line "TO" is perpendicular to line "OP", where "O" is the origin. Then "tangent" specifically also means the length of "PT", and the subtangent is the line "TO", or -- interchangeably -- the length of line "TO".

The subtangent can be found to be

:$TO = - \left\{ ho^2 over ho\text{'}\right\}.$

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### Look at other dictionaries:

• Subtangent — Sub*tan gent, n. (Geom.) The part of the axis contained between the ordinate and tangent drawn to the same point in a curve. [1913 Webster] …   The Collaborative International Dictionary of English

• subtangent — [sub tan′jənt] n. Geom. the segment of the x axis included between the ordinate of a given point on a curve and the tangent at that point …   English World dictionary

• subtangent — /sub tan jeuhnt/, n. Geom. the part of the x axis cut off between the ordinate of a given point of a curve and the tangent at that point. [1705 15; SUB + TANGENT] * * * …   Universalium

• subtangent — sub·tangent …   English syllables

• subtangent — |səb+ noun Etymology: sub + tangent : the projection on the x axis of the portion of the tangent to a curve between the x axis and the point of tangency …   Useful english dictionary

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