Euler, Goldbach and Exact Values of Trigonometric Functions. Hieu D. Nguyen and Elizabeth Volz Rowan University Glassboro, NJ

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1 I. Itoductio Eule, Goldbach ad Exact Values of Tigoometic Fuctios Hieu D. Nguye ad Elizabeth Volz Rowa Uivesity Glassboo, NJ 0808 Jauay 9, 04 I a potio of a lette set to Chistia Goldbach o Decembe 9, 74, Leoad Eule wites (see [8]): I have lately also foud a emakable paadox. Namely that the value of the + + expessio is appoximately equal to 0/3 ad that this factio diffes oly i pats pe millio fom the tuth. The tue value of this expessio howeve is the cosie of the ac o the ac of 39 degees 4 mi. 5 sec. 5 teths of sec. ad 9 hudedths of sec. i a cicle of adius oe. This paadox ca be see moe clealy if we compae umeically the two quatities metioed by Eule (i mode otatio): i + i = cos(l) = = Of couse, Eule s paadox ca be esolved by cosideig the cotiued factio expasio of cos(l) = [0;,3,,,76,,...] o i factio fom: cos(l) = It is well kow that tucatios of a cotiued factio α, called covegets (o patial quotiets), povide atioal appoximatios of α, ad if a coveget pecedes a lage quotiet i the cotiued factio, the it gives a close appoximatio (see Kuschev [5] fo a itoductio to cotiued factios based o Eule s poit of view). This explais Eule s (.)

2 paadox sice 0/3 is the fifth coveget pecedig the quotiet /76 i the cotiued factio (.) : 0 3 = O Febuay 3, 74, Goldbach eplies to Eule ad poses the followig poblem i elatio to the paadox aised i Eule s lette: + + With the obsevatio as it was commuicated to me that is appoximately equal to 0/3 I have oticed that if you wated to make it so that p p + = 0 the p would have to be smalle tha 3 ad lage tha. I cofess that these limits ae lage but I do ot have the cuiosity to detemie them ay close. Eule s eplies back to Goldbach o Mach 6, 74, povidig him with the exact solutio fo p: p Now that I have the cuiosity to ivestigate whe p + = 0 it has give me the oppotuity to emak that such a ifiite mode could happe. Fist obseved that p is betwee ad 3, amely The tue value is π p = wheeπ = ad l= etc. = All l followig values ae deived out of this i that you multiply these with 3,5,7,9 etc. Eule ad Goldbach would exchage fou moe lettes o this topic with potios i each lette discussig thei ow appoaches ad geealizatios to the above poblem, although they eve esolve Eule s paadox. What makes these lettes iteestig, besides the mathematics cotaied i them, is that they seem to idicate Eule s ealiest applicatio of his famous fomula ix e = cos x+ isi x, (.) which was fist published by Eule i 748 i his pe-calculus textbook, Itoductio i aalysi ifiitoum [3]. Although (.) eve appeas explicitly i Eule s lettes, it is ot fafetched to coclude fom his use of the fomula ( ix ix a + a )/= cos( xl a) that Eule cetaily kew of (.) as ealy as 74 (if ot ealie). Moeove, i eadig these lettes oe ecogizes may of Eule s tademak techiques fo exploig ad geealizig mathematical poblems. He is clealy cosideed the maste i compaiso to Goldbach, asweig all of Goldbach s questios at depth ad eve poitig out some of Goldbach s mathematical mistakes. Eule most likely deived 0/3 i this mae sice he had peviously developed the mode theoy of cotiued factios i his wok De Factiolous Cotiious published i 737. Roge Cotes had discoveed the ivese fomula log(cos x + isi x) = ix i 74 (see

3 I this pape we explai the mathematics stemmig fom Eule ad Goldbach s i i cosideatio of the equatio + = 0 ad show how it leads to coectios with cetai exact values of tigoometic fuctios i tems of Fiboacci-Lucas sequeces. I paticula, give a + a = b, Eule claims that fo ay eal value, pi pi b+ b 4 b b 4 a + a = + (.3) Followig up o Eule s esult, if we ow defie pi pi x = a + a = cos( p log a) (.4) fo all positive iteges, the x satisfies the secod-ode liea ecuece x = bx x, (.5) + + with x 0 = 0 ad x = b. Thus, (.4) ad (.5) povide us with a ecusive fomula fo calculatig the values cos( p log a ). We metio that aalogous fomulas ivolvig hypebolic tigoometic fuctios wee deived by T. Osle i [6] ad [7]. II. Fiboacci Sequece ad Biet s Fomula Recall that it is Goldbach who iitiates the poblem of fidig solutios to p p + = 0. Sice cos( p) = ( + ) /, it follows that cos( pi ) = 0, which foces + p = π, whee is a itege. This is the solutio give by Eule i his eply to Goldbach. Goldbach ext cosides solutios to the equatio p p + = 3 (.6) ad without explicitly metioig the solutio fo p, he states the fomula below without poof: x+ x+ x x xp xp (+ 5) ( + 5) (+ 5) ( + 5) + =. (.7) x+ x This fomula Goldbach cosides emakable ad it uclea how he obtais it o why he cosides it. Oe way to deive (.7) is to solve (.6) fo use algeba to maipulate the expessio xp xp 3+ 5 x 3 5 x + = ( ) + ( ) p, which equals 3 ± 5, ad the x+ x+ x x (+ 5) ( + 5) (+ 5) ( + 5) = x+ x Obseve that the left had side of (.7) esembles Biet s fomula fo Fiboacci umbes F : F, (.8) 5 3

4 whee F satisfies the ecuece F+ = F+ + F with F 0 = 0 ad F =. Ideed, the followig idetities hold fo ay itege : + + (+ 5) ( + 5) F = F, + + (.9) (+ 5) ( + 5) F = F. It follows fom addig the two equatios i (.9) ad equatig with (.7) that x = + = F + F, p p + i.e., each x is a sum of Fiboacci umbes (o a bisectio of Lucas umbes) 3. Hee ae the fist few values of x : x = F + F = + =, x = F + F = + = x = F + F = 5+ = 7, x = F + F = 3+ 5= We ote that Eule eve metios to Goldbach the coectio betwee (.7) ad the Fiboacci umbes. This is supisig give the ecusive atue of the Fiboacci umbes ad the cicumstaces suggestig that Eule kew of fomula (.8) befoe the. I paticula, Daiel Beoulli had published fomula (.8) i 78 i [] (Sectio 7) ad Beoulli is kow to have had may coespodeces with Eule duig the peiod Eule himself published a vaiatio of (.8), but much late i 748, i [3] ad well befoe Biet s idepedetly discovey of it i 843 i []. I fact, De Moive seems to be the fist peso to have discoveed this fomula. I his 7 pape [4], he implicitly deives the followig well-kow powe seies expasio fo the ecipocal of x x by patial factio decompositio: = F + x. x x = 0 The bidge coectig the two sides of this equatio is of couse (.8). II. Liea Recueces ad Special Values of Tigoometic Fuctios I espose to Goldbach s solutio of the equatio moe geeal situatio: If a + a = b, the pi pi b+ b 4 b b 4 a + a = + This follows fom solvig the quadatic equatio a + a = b fo p p 3 + =, Eule cosides the pi a : pi b± b 4 a = pi pi If we defie x = a + a, whee is a o-egative itege, the x satisfies the liea ecuece 3 Sequece A00548 i The Olie Ecyclopedia of Itege Sequeces: 4

5 x = bx x + + x =, x = b 0 To pove this, we fist show that the elemetay solutio 5, y (.0) pi = a satisfies the same ecuece. ( ) pi This follows fom multiplyig the idetity a + a = b by a + to obtai ( + ) pi ( + ) pi pi a = ba a, o equivaletly, y = by y. (.) + + pi a pi It is easy to check that the othe elemetay solutio also satisfies the same ecuece. By pi lieaity, the sequece x = a + a satisfies the ecuece as well. Thus, b+ b 4 b b 4 x = + Sice x ca be witte i the tigoometic fom pi pi x = a + a = cos[ pl a], we obtai as a esult the followig fomula fo special values of cosie: b+ b 4 b b 4 cos[ p l a] + (.) It is ot difficult to show that a coespodig fomula holds fo the sie fuctio as well. I paticula, if the equatio a a = b holds, the pi pi b b b b a + ( ) a = + ( ). We claim that x = a pi + ( ) a pi satisfies the followig ecuece fo itege values of : x = bx + x + + x = 0, x = b 0 which is aalogous to (.0). This follows fom the fact that, y (.3) pi = a ad z ( ) pi a = ae both elemetay solutios of the same ecuece. Fo y, this ca be pove usig the same ( ) agumet as that used to establish (.). Fo z, we multiply a a = b by ( ) + a + pi to obtai + pi + ( + ) pi + ( + ) pi ( ) a ( ) a = b( ) a, o equivaletly, z = bz + z. + + Thus, by lieaity the geeal solutio x = y + z satisfies the same ecuece as well, which we wite i the tigoometic fom pi pi cos[ p l a] if eve x = a + ( ) a = si[ p l a] if odd This esults i the followig fomula fo special values of sie fo a odd itege:

6 b+ b + 4 b b + 4 si[ p l a] +. (.4) We leave it fo the eade to show that fo a eve itege, b+ b + 4 b b + 4 si[ p l a]. (.5) We coclude by obsevig that Osle s hypebolic vesio of fomulas (.4) ad (.5) deived pi i [6] ca be obtaied by eplacig the quatities a ad b by a ad M, espectively. REFERENCES. J. P. Biet, Mémoie su l itégatio des équatios liéaies aux difféeces fiies, d u ode quelcoque, à coefficiets vaiables. Comptes Redus hebdomadaies des séaces de l Académie des Scieces (Pais), 7 (843), D. Beoulli, Obsevatioes de seiesbus quae fomatu ex additioe vel subtactio quacuque temioum se mutuo cosequetium, ubi paesetim eaudem isigis usus po iveiedid adic[ibus] omium Aequatioum Algebaicaum osteditu, Commetaii Academiae Scietiaum Impeialis Petopolitaae, 3 (78), (Icoectly titled Obsevatioes de Seiebus Recuetibus by Rudio i the Opeia Omia) 3. L. Eule, Itoductio i aalysi ifiitoum [E0], 748, Chapte XIII, Example 3, p. 84. Available at The Eule Achive: 4. A. de Moive, De Factioibus Algebaicis Radicalitate Immuibus ad Factioes Simplicioes Reducedis, Deque Summadis Temiis Quaumdam Seieum Aequali Itevallo a Se Distatibus, Philos. Tas., 3 (7), S. Khuschchev, Othogoal Polyomials ad Cotiued Factios: Fom Eule s Poit of View, Cambidge U. Pess, New Yok, T. Osle. Exact values of the hypebolic fuctios, Mathematics ad Compute Educatio, 4 (008), T. Osle. Moe exact values of the hypebolic fuctios, Mathematics ad Compute Educatio, 4 (008), Volz, Elizabeth. A Eglish taslatio of potios of seve lettes betwee Eule ad Goldbach o Eule s complex expoetial paadox ad special values of cosie, 008. Available at: os/eule%0goldbach%0lettes%0complex%0expoetial%0paadox%0eglis h%0taslatio.pdf 6