# 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 *θ* be the angle of inclination of the tangent with respect to the *x*-axis. Let the curve be described by *y=f(x)*, let *x _{0}* be the abscissa of point

*P*, and let

*θ*be the angle of inclination of the tangent of

_{0}*P*. Then this tangent of

*P*is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t = f(x_0) \, \csc \theta_0 \quad }**

and the subtangent is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_s = t \, \cos \theta_0 = f(x_0) \cot \theta_0 \quad }**

The angle of inclination *θ* is related to the derivative by

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta = \arctan {df \over dx} }**

therefore

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_s = { f(x_0) \over f'(x_0) }. }**

## 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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle TO = - {\rho^2 \over \rho'}. }**