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 the circle.
The exsecant,^{[2]}^{[3]}^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]} (Latin: secans exterior^{[10]}^{[11]}^{[12]}^{[13]}) also known as exterior, external,^{[14]}^{[15]}^{[16]}^{[17]}outward or outer secant and abbreviated as exsec^{[18]}^{[19]}^{[20]}^{[21]} or exs,^{[22]} is a trigonometric function defined in terms of the secant function sec(θ):^{[23]}
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 lineOE, and the exsecant is the portion DE of this secant that lies exterior to the circle (ex is Latin for out of).
ExcosecantEdit
exsecant (blue) and excosecant (green)
A related function is the excosecant^{[5]}^{[24]} or coexsecant,^{[25]}^{[18]}^{[26]} also known as exterior, external,^{[17]}outward or outer cosecant and abbreviated as excosec, coexsec,^{[14]}^{[18]}^{[26]}excsc^{[5]}^{[24]} or exc,^{[22]} the exsecant of the complementary angle:^{[24]}
The reason to define a special function for the exsecant is similar to the rationale for the versine: for small anglesθ, the sec(θ) function approaches one, and so using the above formula for the exsecant will involve the subtraction of two nearly equal quantities, resulting in catastrophic cancellation. 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, if using the cosine-based definition. A more accurate formula in this limit would be to use the identity:^{[3]}^{[4]}^{[17]}
Prior to the availability of computers, this would require time-consuming multiplications.
The exsecant function was used by Galileo Galilei in 1632 already, although he still called it segante (meaning secant).^{[27]}^{[28]}^{[29]}^{[30]} The Latin term secans exterior was used since at least around 1745.^{[10]}^{[11]}^{[12]}^{[13]} The usage of the English term external secant and the abbreviation ex. sec. can be traced back to 1855 the least, when Charles Haslett published the first known table of exsecants.^{[1]}^{[31]} Variations such as ex secant and exsec were in use in 1880,^{[14]} and exsecant was used since 1894 the least.^{[2]}
The terms coexsecant^{[25]} and coexsec^{[2]} can be found used as early as 1880 as well^{[2]}^{[25]} followed by excosecant since 1909.^{[5]} The function was also utilized by Albert Einstein to describe the kinetic energy of fermions.^{[29]}^{[30]}
The inverse functions arcexsecant^{[26]} (arcexsec,^{[5]}^{[26]}aexsec,^{[32]}^{[33]}aexs, exsec^{−1}) and arcexcosecant (arcexcosec, arcexcsc,^{[5]}aexcsc, aexc, arccoexsecant, arccoexsec, excsc^{−1}) exist as well:
$\operatorname {arcexsec} (y)=\operatorname {arcsec}(y+1)=\arccos \left({\frac {1}{y+1}}\right)=\arctan({\sqrt {y^{2}+2y}})$^{[26]}^{[32]}^{[33]} (for y ≤ −2 or y ≥ 0)^{[26]}
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