Euler's Opuscula Analytica Vol. I : How sines and cosines of multiple angles. [E562]. HOW THE SINES AND COSINES OF MULTIPLE ANGLES

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1 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 1 HOW THE SINES AND COSINES OF MULTIPLE ANGLES MAY BE ABLE TO BE EXPRESSED BY PRODUCTS Opuscula aalytica p [E56] 1. For some agle proposed there may be put for the sake of brevity ad there will be the truly from which there becomes ad p cos. 1si. q cos. 1si. pq 1; p cos. 1si. ad q cos. l si. p q cos. p q 1si.. Therefore the matter is reduced to that so that the formulas resolved ito factors. p q ad p q may be. Iitially we will cosider the formula p q cos. which as ofte as is a odd umber it has the simple factor p q cos. thus so that i these cases cos. shall be a factor of cos.. But for the remaiig cases we may put the twofold factor i geeral to be pp pqcos. qq thus so that the formula p q may vaish o puttig but the there will be either or pp pqcos. qq 0; pq(cos. 1 si. ) pq(cos. 1 si. )

2 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. ad hece p q (cos. 1 si. ) ; ad thus there will have to be q (cos. 1si. ) q 0 or cos. 1si. 1 0 from which there becomes fit si. 0 ad cos. 1 but the there becomes immediatelysi Therefore sice cos. 1 the agle will be either etc. Ad thus if i may deote some odd umber there will be i ad hece i o accout of which the twofold factor i geeral will be 4. Now sice there shall be i pppqcos. qq pp qq cos. o accout of pq 1there will be this factor cos. cos. i which is resolved at oce ito two factors. Ideed sice there shall be there will be B A B cos. Acos. B si. si. A i i i cos. cos. si.( )si.( ) ad thus oe sigle factor i geeral will be i i 4si.( )si.( ). Hece by writig for i successively the umbers etc. there will be cos. 4si.( )si.( ) si.( )si.( ) 4si.( )si.( ) etc. the altogether there may be had factors.

3 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com Therefore we may ru through this expressio accordig to the sigle values of the umber ad there will be if 1 cos. si.( ) if cos. si.( )si.( ) if 3 cos.3 si.( )si.( )si.( ) if 4 cos. 4 si.( )si.( )si.( )si.( ) if 5 cos. 5 si.( )si.( )si.( )si.( )si.( ) if 6 cos. 6 si.( )si.( )si.( )si.( )si.( )si.( ). Moreover geerally there will be cos. si.( )si.( )si.( )si.( ) etc. the there may be had factors. 6. Therefore there will be with the logarithms take : 1 which equatio differetiated will produce 3 l 3 lcos. l lsi.( ) lsi.( ) lsi.( ) si.( ) etc. that is si. d cos.( ) d cos.( d ) cos. si.( ) si.( ) dcos.( 3) dcos.( 3) si.( 3) si.( 3) etc. ta. cot.( ) cot.( ) 3 3 cot.( ) cot.( ) etc. from which the followig oteworthy equatios are deduced :

4 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 4 I. ta. cot.( ) II. ta. cot.( ) cot.( ) ta.( ) ta.( ) III. 3ta.3 cot.( ) cot.( ) cot.( 3 ) or IV. 4ta.4 cot.( ) cot.( ) cot.( 3 ) cot.( 3 ) or 4ta.4 ta.( 3 ) ta.( 3 ) ta.( ) ta.( ) I the same maer we may treat the formula p q 1 si. the twofold factor of which we may put i place : i which o puttig pp pqcos. qq 0 so that there becomes as before ad hece agai: pq(cos. 1 si. ) p q (cos. 1 si. ); ad thus there will have to become: q (cos. 1si. ) q 0 or cos. 1 si. 1 0 from which there must become si. 0 ac cos. 1 o accout of which the agle will be either or i geeral i ; ad thus with i deotig all the umbers etc. Hece therefore the twofold factor will be i be geeral: pp pqcos. i qq cos. cos. i i

5 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 5 which is resolved ito these factors i i si.( ) si.( ); but besides the formula p q has the simple factor ad cosequetly we will have pq 1 si. ; ad thus i i si. si. si.( ) si.( ) etc. si. si. si.( ) si.( ) si.( ) si.( ) etc. the overall factors may be produced. Therefore there will be 1 si. si. si.( )si.( ) si.( ) si.( ) etc. 8. Now from this geeral form we may deduced the followig special forms : if 1 si. si.( ) 0 if si. si.( )si.( ) ( ) if 3 si.3 4si.( )si.( )si.( ) if 4 si.4 8si.( )si.( )si.( ) if 5 si.5 16si.( )si.( )si.( )si if 6 si.6 3si.( )si.( )si.( )si.( )si.( ). 9. Here also as before we may take logarithms ad there will be 1 lsi. l lsi.( ) lsi.( ) etc. which equatio differetiated ad divided by d provides : or cos.( ) cos.( ) cos.( cos. cos. ) si. si. si.( ) si.( ) si.( ) etc.

6 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 6 cot. cot. cot.( ) cot.( ) cot.( ) etc. the we may have terms. 10. Hece therefore we will have the followig special forms: if 1 cot. cot. if cot. cot. cot.( ) if 3 3cot.3 cot. cot.( ) cot.( ) if 4 4cot.4 cot. cot.( ) cot.( ) cot.( ). 11. If we may differetiate aew the formula foud for cot. o accout of o dividig by d we will have d cot. d si si. si. si.( ) si.( ) si.( ) etc. the we may have terms from which the followig cases may be observed : 1 1 si. si. if si. si. si.( ) if si.3 si. si.( ) si.( ) 3 3 if si.4 si. si.( ) si.( ) si.( ) if 4. THE EVOLUTION OF THE FORMULA p pq cos. q 1. We may assume here as before p cos. 1si. et q cos. 1si. thus so that this formula may ivolve this value

7 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 7 cos. cos. 4si.( 1)si.( 1 ). Now pppqcos. qq shall be a twofold factor of this formula which therefore must vaish o puttig pq(cos. 1 si. ) from which with the factor substitutes there will be produced q (cos. 1si. ) q cos. (cos. 1si. ) q 0 that is cos.cos. cos. 1 1si. cos. 1si. 0 from which these two equatios arise ad cos. cos. cos. 1 0 si. cos. si. 0. Now sice there shall be these two equatios will be cos. cos. 1 ad si. si. cos. or cos. cos. cos. 0 ad si. cos. cos. si. 0 cos. cos. 0 ad cos. cos. 0 from which it follows cos. cos.. Therefore there will be either 4 6 or i geeral i from which there becomes i geeral i thus so that i may deote the umbers etc. 13. Therefore i geeral the formulas of our twofold factor will be Truly there is cos. i pp qq pq. pp qq cos. ad pq 1

8 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 8 from which this factor will be which is reduced to these simple factors i cos. i i 4si. si. ; from which i place of i by writig the umbers etc. the factors of our formulas will be si. si. 4si. si. 4si. si. 4si. si. etc. which factors must be cotiued to that poit at which the umber of these may become. 14. Therefore sice this product shall be equal to the formula 1 1 4si.( )4si.( ) ad i our product the umerical factor shall be o dividig by 4 we will have this equatio si.( )si.( ) si. si. si. si. si. si. etc. which equatio so that it may be retured more cocisely we may put ad there will be si. ( )si. ( ) si.( )si.( ) si.( )si.( ) si.( )si.( ) etc. 15. But this equatio is ot ew but ow may be cotaied i the precedig oe which was 1 si. si. si.( )si.( )si.( )si.( ) etc. ad sice there shall be there will be si.( o) si.( o )

9 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 9 ( 1) si.( ) si.( ) ( ) si.( ) si.( ) 3 ( 3) si.( ) si.( ); from which that expressio may be reduced to this form 1 3 ( 1) si.( ) si.( ) si. si. si.( )si.( ) where the arcs are cotiued i a arithmetical progressio. So that if ow here i place of we may write iitially the hece the two followig formulas may arise: 1 si. ( ) si.( )si.( ) 3 si.( )si.( ) etc. 1 si. ( ) si.( )si.( ) 3 si.( )si.( ) etc. which two equatios multiplied together preset 3 ) etc. si. ( )si. ( ) si.( )si.( ) si.( )si.( ) si.( )si.( ) si.( 16. If ow we may atted to the origi of our formulas sice from our formula this has arise with ad beig preset if we may put p pq cos. q 4 si. ( )si. ( ) p cos. 1si. q cos. 1si.

10 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 10 ad the there will be Thece if we put ad f cos.( ) 1 si.( ) g cos. ( ) 1si.( ). f g 1si. ( ). h cos.( ) 1 si.( ) k cos. ( ) 1si.( ) i a like maer there will be h k 1si. ( ) from which there will become ( f g )( h k ) 4si. ( )si. ( ) p pq cos. q. For this requirig to be demostrated it is to be observed from which there becomes f p(cos. 1 si. ) g q(cos. 1si. ) hq(cos. 1si. ) k p(cos. 1si. ) f p (cos. 1 si. ) g q (cos. 1si. ) h q (cos. 1 si. ) k p (cos. 1 si. ). We may put for brevity therefore cos. 1si. A cos. 1si. B so that there shall be

11 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 11 f Ap g Bq h Aq et k Bp ad hece agai: ad f g Ap Bq h k Aq Bp which two formulas multiplied preset where sice there shall be ( f g )( h k ) ( A B ) p q AB( p q ); this product will be AB 1 ad AA BB cos. p pq cos. q which is that itself which we have foud. COROLLARY Hece therefore we uderstad the formula to be resolved ito these two factors p pq cos. q with ( Ap Bq ) ad ( Bp Aq ) Acos. a 1 si. B cos. a 1 si.. beig preset.

12 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 1 QUOMODO SINUS ET COSINUS ANGULORUM MULTIPLORUM PER PRODUCTA EXPRIMI QUEANT Commetatio 56 idicis ENESTROEMlANI Opuscula aalytica p Proposito agulo quocuque poatur brevitatis gratia erit tum vero ude fit Res igitur eo redit ut formulae p cos. 1 si. et q cos. 1 si. pq 1; p cos. 1si. et q cos. l si. p q cos. et p q 1si.. p q et p q i factores resolvatur.. Cosideremus primo formulam p q cos. quae quoties est umerus impar factorem habet simplicem p q cos. ita ut his casibus cos. sit factor ipsius cos.. Pro reliquis factoribus autem poamus factorem duplicem i geere esse pppqcos. qq ita ut formula p q evaescat posito tum autem erit vel pp pqcos. qq 0; pq(cos. 1 si. ) vel hicque pq(cos. 1 si. )

13 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 13 p q (cos. 1 si. ) ; sicque debebit esse sive q (cos. 1si. ) q 0 cos. 1si. 1 0 ude fit si. 0 et cos. 1 tum autem spote fit si Quia igitur cos. 1 agulus erit vel vel 3 vel 5 vel 7 vel etc. Sicque si i deotet umerum imparem quemcuque erit i hicque i quocirca factor duplex i geere erit pppqcos. i qq 4. Cum uc sit pp qq cos. ob pq 1erit iste factor cos. cos. i qui spote i duos factores resolvitur. Cum eim sit cos. Acos. B si. BAsi. BA erit cos. cos. i si.( i )si.( i ) sicque uus factor i geere erit i i 4si.( )si.( ). Hic pro i successive umeros etc. scribedo erit cos. 4si.( )si.( ) si.( )si.( ) 4si.( )si.( ) etc. doec omio habeatur factores. 5. Percurramus igitur hac expressioem secudum sigulos valores umeri eritque

14 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com si.( )si.( )si.( 3 )si.( 3 )si.( 5 ) si 1 cos. si.( ) si cos. si.( )si.( ) si 3 cos.3 si.( )si.( )si.( ) si 4 cos. 4 si.( )si.( )si.( )si.( ) si 5 cos. 5 si 6 cos. 6 si.( )si.( )si.( )si.( )si.( )si.( ). Geeraliter autem erit cos. si.( )si.( )si.( )si.( ) etc. doec habeatur factores. 6. Sumedis igitur logarithmis erit quae aequatio differetiata praebet 1 3 l 3 lcos. l lsi.( ) lsi.( ) lsi.( ) si.( ) etc. hoc est si. d cos.( ) d cos.( d ) cos. si.( ) si.( ) dcos.( 3) dcos.( 3) si.( 3) si.( 3) etc. ta. cot.( ) cot.( ) 3 3 cot.( ) cot.( ) etc. ude deducutur sequetes aequalitates memoratu digae

15 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 15 I. ta. cot.( ) II. ta. cot.( ) cot.( ) ta.( ) ta.( ) III. 3ta.3 cot.( ) cot.( ) cot.( 3 ) sive IV. 4ta.4 cot.( ) cot.( ) cot.( 3 ) cot.( 3 ) sive 4ta. 4 ta.( 3 ) ta.( 3 ) ta.( ) ta.( ). 7. Eodem modo tractemus formulam cuius factorem duplicem statuamus p q 1 si. quo posito 0 fit ut ate hicque porro sicque debebit esse sive ude fieri debet pp pqcos. qq pq(cos. 1 si. ) p q (cos. 1 si. ); q (cos. 1si. ) q 0 cos. 1 si. 1 0 si. 0 ac cos. 1 quamobrem agulus erit vel 0 vel vel 4 vel 6 vel i geere i ideoque deotate i umeros omes etc. Hic igitur factor duplex i geere erit qui resolvitur i hos factores i i i pp pqcos. qq cos. cos.

16 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 16 si.( i ) si.( i ); praeterea autem formula cosequeter habebimus p q habet factorem simplicem pq 1 si. ; ideoque i i si. si. si.( ) si.( ) etc. si. si. si.( ) si.( ) si.( ) si.( ) etc. doec omio prodeat factores. Erit ergo 1 si. si. si.( )si.( ) si.( ) si.( ) etc. 8. Iam ex hac forma geerali sequetes deducamus formas speciales: si 1 si. si.( ) 0 si si. si.( )si.( ) ( ) si 3 si.3 4si.( )si.( )si.( ) si 4 si.4 8si.( )si.( )si.( ) si 5 si.5 16si.( )si.( )si.( )si si 6 si.6 3si.( )si.( )si.( )si.( )si.( ). 9. Sumamus hic etiam ut ate logarithmos eritque 1 lsi. l lsi.( ) lsi.( ) etc. quae aequatio differetiata et per d divisa praebet sive cos.( ) cos.( ) cos.( cos. cos. ) si. si. si.( ) si.( ) si.( ) etc. cot. cot. cot.( ) cot.( ) cot.( ) etc.

17 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 17 doec habeatur termii. 10. Hic igitur sequetes obtiebimus formas speciales: si 1 cot. cot. si cot. cot. cot.( ) si 3 3cot.3 cot. cot.( ) cot.( ) si 4 4cot.4 cot. cot.( ) cot.( ) cot.( ). 11. Si formulam pro cot. ; ivetam deuo differetiemus ob d cot. d per d dividedo habebimus si si. si. si.( ) si.( ) si.( ) etc. doec habeatur termii ude sequetes casus otetur: 1 1 si. si. si si. si. si.( ) si si.3 si. si.( ) si.( ) 3 3 si si.4 si. si.( ) si.( ) si.( ) si 4. EVOLUTIO FORMULAE p pq cos. q 1. Sumamus hic ut ate p cos. 1 si. et q cos. 1si. ita ut ista formula ivolvat huc valorem 1 1 cos. cos. 4si.( )si.( ).

18 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 18 Iam sit pp pqcos. qq factor duplex huius formulae quae ergo evaescere debet posito pq(cos. 1 si. ) ude facta substitutioe prodibit q (cos. 1si. ) q cos. (cos. 1 si. ) q 0 hoc est cos.cos. cos. 1 1si. cos. 1si. 0 ude ascutur hae duae aequatioes cos. cos. cos. 1 0 et si.cos. si. 0. Cum uc sit cos. cos. 1 et si. si. cos. hae duae aequatioes erut sive cos. cos. cos. 0 et si. cos. cos. si. 0 cos. cos. 0 et cos. cos. 0 ude sequitur cos. cos.. Erit ergo vel vel vel 4 vel 6 vel i geere i ude fit i geere ita ut i deotet umeros etc. i 13. Formulae igitur ostrae factor duplex i geere erit Est vero ude hic factor erit cos. i pp qq pq. pp qq cos. et pq 1 i cos. qui reducitur ad hos factores simplices

19 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 19 i i 4si. si. ; ude loco i scribedo umeros etc. factores ostrae formulae erut si. si. 4si. si. 4si. si. 4si. si. etc. qui factores eousque cotiuari debet doec eorum umerus fiat 14. Cum igitur hoc productum aequale sit formulae 1 1 4si.( )4si.( ). et i ostro producto factor umericus sit hac aequatioem 4 per 4 dividedo habebimus si.( )si.( ) si. si. si. si. si. si. etc. quae aequatio quo cociior reddatur poamus et erit si. ( )si. ( ) si.( )si.( ) si.( )si.( ) si.( )si.( ) etc. 15. Haec autem expressio o est ova sed iam i praecedete cotietur quae erat et cum sit erit 1 si. si. si.( )si.( )si.( )si.( ) etc. si.( o) si.( o ) 3 ( 1) ( ) ( 3) si.( ) si.( ) si.( ) si.( ) si.( ) si.( ); ude illa expressio reducetur ad hac formam

20 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 0 1 si. si. si.( )si.( ) 3 ( 1) si.( ) si.( ) ubi arcus i progressioe arithmetica cotiua progrediutur. Quodsi iam hic loco scribamus primo deide hic duae formulae sequetes ascetur: 1 si. ( ) si.( )si.( ) 3 si.( )si.( ) etc. 1 si. ( ) si.( )si.( ) 3 si.( )si.( ) etc. quae duae aequatioes i se ivicem ductae praebet 3 ) etc. si. ( )si. ( ) si.( )si.( ) si.( )si.( ) si.( )si.( ) si.( 16. Si uc attedamus ad origiem harum formularum quadoquidem ex ostra formula ata est haec p pq cos. q existete si poamus 4 si. ( )si. ( ) p cos. 1 si. et q cos. 1si. f cos.( ) 1si.( ) et g cos. ( ) 1si.( ). tum erit

21 Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. 1 Deide si poamus erit simili modo ude erit f g 1si. ( ). h cos.( ) 1si.( ) et k cos. ( ) 1 si.( ) h k 1si. ( ) ( f g )( h k ) 4si. ( )si. ( ) p pq cos. q.. Ad hoc demostradum otetur esse f p(cos. 1 si. ) g q(cos. 1si. ) hq(cos. 1si. ) k p(cos. 1si. ) ude fit f p (cos. 1 si. ) Poamus brevitatis ergo g q (cos. 1si. ) h q (cos. 1si. ) k p (cos. 1si. ). ut sit cos. 1si. A cos. 1si. B f Ap g Bq h Aq et k Bp hicque porro f g Ap Bq et h k Aq Bp quae duae formulae multiplicatae praebet

22 ubi cum sit Euler's Opuscula Aalytica Vol. I : How sies ad cosies of multiple agles. [E56]. Tr. by Ia Bruce : September 017: Free Dowload at 17ceturymaths.com. ( f g )( h k ) ( A B ) p q AB( p q ); hoc productum erit AB 1 et AA BB cos. p pq cos. q quod est id ipsum quod iveimus. Hic igitur itelligimus formulam COROLLARIUM resolvi i hos duos factores existete p pq cos. q ( Ap Bq ) et ( Bp Aq ) Acos. a 1 si. B cos. a 1 si..