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