# Exsecant

exsecant (blue) and excosecant (green)
The trigonometric functions, including the exsecant, can be constructed geometrically in terms of a unit circle centered at O. The exsecant is the portion DE of the secant exterior to (ex) the circle.

The exsecant, also abbreviated exsec, is a trigonometric function defined in terms of the secant function sec(θ):

$\operatorname{exsec}(\theta) = \sec(\theta) - 1. \,$

Once important in fields such as surveying, astronomy, and spherical trigonometry, the exsecant function is now little-used. Mainly, this is because the availability of calculators and computers has removed the need for trigonometric tables of specialized functions such as this one.

A related function is the excosecant (excsc), the exsecant of the complementary angle:

$\operatorname{excsc}(\theta) = \operatorname{exsec}(\pi/2 - \theta) = \csc(\theta) - 1. \!$

The reason to define a special function for the exsecant is similar to the rationale for the versine: for small angles θ, the sec(θ) function approaches one, and so using the above formula for the exsecant will involve the subtraction of two nearly equal quantities and exacerbate roundoff errors. Thus, a table of the secant function would need a very high accuracy to be used for the exsecant, making a specialized exsecant table useful. Even with a computer, floating point errors can be problematic for exsecants of small angles. A more accurate formula in this limit would be to use the identity:

$\operatorname{exsec}(\theta) = \frac{1-\cos(\theta)}{\cos(\theta)} = \frac{\operatorname{versin}(\theta)}{\cos(\theta)} = 2 \sin^2(\theta/2) \sec(\theta).\$

Prior to the availability of computers, this would require time-consuming multiplications.

The name exsecant can be understood from a graphical construction, at right, of the various trigonometric functions from a unit circle, such as was used historically. sec(θ) is the secant $\overline{OE}$, and the exsecant is the portion $\overline{DE}$ of this secant that lies exterior to the circle (ex is Latin for out of).