Exsecant

The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric functions defined in terms of the secant and cosecant functions. They were important in fields such as surveying, railway engineering, civil engineering, astronomy, and spherical trigonometry and could help improve accuracy, but are rarely used today except to simplify some calculations.

A unit circle with trigonometric functions.

Exsecant

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 known as exterior, external, outward or outer secant and abbreviated as exsec or exs, is a trigonometric function defined in terms of the secant function sec(θ):

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

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

Excosecant

File:Exsecant and exosecant plot.png

exsecant (blue) and excosecant (green)

A related function is the excosecant or coexsecant, also known as exterior, external, outward or outer cosecant and abbreviated as excosec, coexsec, excsc or exc, the exsecant of the complementary angle:

$\operatorname{excsc}(\theta) = \operatorname{exsec}\left(\frac{\pi}{2} - \theta\right) = \csc(\theta) - 1 = \frac{1}{\sin(\theta)} - 1.$

Usage

Once important in fields such as surveying, railway engineering (for example to lay out railroad curves and superelevation), civil engineering, 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.

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 round-off 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:

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

or

$\operatorname{excsc}(\theta) = \frac{1-\sin(\theta)}{\sin(\theta)} = \frac{\operatorname{coversin}(\theta)}{\sin(\theta)} = \operatorname{coversin}(\theta) \csc(\theta) = 2 \left(\cos\left(\frac{\theta}{2}\right)\right)^2 \csc(\theta).\$

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

Mathematical identities

Derivatives

$\frac{\mathrm{d}}{\mathrm{d}\theta}\operatorname{exsec}(\theta) = \tan(\theta)\sec(\theta) = \frac{\sin(\theta)}{(\cos (\theta))^2}$

$\frac{\mathrm{d}}{\mathrm{d}\theta}\operatorname{excsc}(\theta) = -\cot(\theta)\csc(\theta) = \frac{-\cos(\theta)}{(\sin(\theta))^2}$

Integrals

$\int\operatorname{exsec}(\theta)\,\mathrm{d}\theta = \ln\left[\cos\left(\frac{\theta}{2}\right) + \sin\left(\frac{\theta}{2}\right)\right] - \ln\left[\cos\left(\frac{\theta}{2}\right) - \sin\left(\frac{\theta}{2}\right)\right] - \theta + C$

Inverse functions

The inverse functions arcexsecant (arcexsec, aexsec, aexs, exsec⁻¹) and arcexcosecant (arcexcosec, arcexcsc, aexcsc, aexc, excsc⁻¹) exist as well:

$\operatorname{arcexsec}(y) = \arcsec(y+1) = \arccos\left(\frac{1}{y+1}\right) = \arctan(\sqrt{y^2+2y})$ (for y ≤ −2 or y ≥ 0)

$\operatorname{arcexcsc}(y) = \arccsc(y+1) = \arcsin\left(\frac{1}{y+1}\right)\,$

Other properties

The exsecant and excosecant functions can be extended into the complex plane.

$\operatorname{exsec}(\theta) = \tan(\theta) \tan\left(\frac{\theta}{2}\right)$

See also

• Trigonometric identities
• Versine and haversine
• Chord
• Exponential minus 1
• Natural logarithm plus 1