Some Quaint & Curious & Almost Forgotten Trig Functions #80 of Gottschalk’s Gestalts A Series Illustrating Innovative Forms of the Organization & Exposition of Mathematics by Walter Gottschalk Infinite Vistas Press PVD RI 2002 GG80-1 (25) „ 2002 Walter Gottschalk 500 Angell St #414 Providence RI 02906 permission is granted without charge to reproduce & distribute this item at cost for educational purposes; attribution requested; no warranty of infallibility is posited GG80-2 six related trigonometric functions, antiquated & quaint & curious, that are primarily of historical interest, with some still sometimes somewhat useful let • A Œ angle then • the versed sine of A =ab the versine of A =dn vers A =rd ver - sine A = verse A =df 1- cos A wh versine ¨ versed sine vers ¨ vers ine versed = turned GG80-3 • the coversed sine of A = the versed cosine of A =ab the coversine of A =dn covers A =rd koh - ver - sine A = koh - verse A =df 1 - sin A wh coversed sine ¨ versed sine of complement coversine ¨ coversed sine covers ¨ coversine note that the versed sine of A = 1- cosA & the versed cosine of A = 1 - sin A GG80-4 • half of the versed sine of A = half of the versine of A =ab haversine of A =dn havers A = hav A =rd hav - er - sine A = hav - erse A = have A 1 = vers A df 2 1 = 1 - cos A 2 ( ) wh haversine ¨ half of versine havers ¨ havers ine hav ¨ haversine GG80-5 • half of the coversed sine of A = half of the versed cosine of A = half of the coversine of A =ab hacoversine of A =dn hacovers A =rd hack - o - ver - sine A = hack - o - verse A 1 = covers A df 2 1 = 1 - sin A 2 ( ) wh hacoversine ¨ half of coversine hacovers ¨ hacoversine GG80-6 • the external secant of A =ab the exsecant of A =dn exsec A =rd ecks - see - cant A = ecks - seck A =df sec A- 1 wh exsecant ¨ external secant exsec ¨ exsecant GG80-7 • the external cosecant of A =ab the excosecant of A =dn excsc A =rd ecks - koh - see - cant A = ecks - koh - seck A =df csc A - 1 wh excosecant ¨ external cosecant excsc ¨ excosecant GG80-8 some identities involving these trig fcns 2 A • vers A = 1- cosA = 2 sin 2 • covers A = 1 - sin A 1 2 A • havers A = 1- cos A = sin 2 ()2 1 • hacovers A = 1 - sin A 2 () • exsec A = sec A -1 • excsc A = cscA -1 GG80-9 • vers A = 2havers A • covers A = 2covers A 1 • havers A = vers A 2 1 • hacovers A = covers A 2 • exsec A = sec A vers A • excsc A = csc A covers A GG80-10 • vers Aˆ = covers A • covers Aˆ = vers A • havers Aˆ = hacovers A • hacovers Aˆ = havers A • exsec Aˆ = excsc A • excsc Aˆ = exsec A wh Aˆ =rd comp A = A comp =df the complement of A comp ¨ complement note the overscript suggests a right angle opening downward GG80-11 • vers A = 2 - vers A • covers A = covers A • havers A = 1 - havers A • hacovers A = hacovers A • exsec A = --2 exsec A • excsc A = excosec A wh A =rd sup A = A sup =df the supplement of A sup ¨ supplement note the overscript suggests a straight angle GG80-12 •) vers (-A = vers A • covers(-- A) = 2 covers A •) havers (-A = havers A • hacovers(-- A) = 1ha covers A •) exsec(-A = exsec A • excsc(---AA) = 2 excsc GG80-13 • vers A≥ 0 • covers A≥ 0 • havers A≥ 0 • hacovers A≥ 0 G80-14 the haversine formula for the angles of a plane triangle sbsc • hav A = ()- ()- bc & cyclically the haversine formula for the angles of a spherical triangle • hav A sin()sbsin( sc ) = -- sinbc sin hav a hav() b c = -- sinbc sin = hav[]p -+() B C+ sin B sin C hav a & cyclically GG80-15 the haversine formula for the sides of a spherical triangle • hav a = hav() b - cbc+ sin sin hav A & cyclically note: this formula may be used to find the great - circle distance and the bearing between two positions on the Earth's surface once their latitude & longitude are known GG80-16 in times gone by viz in the 18th & the 19th & the early part of the 20th centuries these trig functions were used rather frequently in geography & in marine navigation; even today you may see some appearance of some of them & not only in matters involving the history of mathematics GG80-17 a study of the following labeled diagrams will reveal reasons for the designations of the four trig functions • the versed sine of A = vers A = 1- cos A • the coversed sine of A = covers A =- 1 sin A • the exsecant of A = exsec A =- sec A 1 • the excosecant of A = excsc A = csc A -1 GG80-18 etymology • sinus rectus (Latin, historical term) = vertical sine = sine • sinus versus (Latin, historical term) = versed sine = sine turned on its side = versine • coversine = versine of complement • exsecant = external part of the secant • excosecant = external part of the cosecant GG80-19 1 sin A A cos A vers A GG80-20 cover s A sin A A 1 GG80-21 exsec A sec A AA 11 GG80-22 A 1 csc A excsc A GG80-23 Aˆ 1 A GG80-24 identify the line segments representing ie whose lengths are ˆ sinAA = cos ˆ cosAA = sin ˆ tanAA = cot ˆ cotAA = tan ˆ secAA = csc cscA = secAˆ versA = coversAˆ covers A = versAˆ exsecA = excscAˆ excscA = exsecAˆ GG80-25.