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Jession9we needtoknowseriesexpansionsof cost andsin Loaf 1,2018easier laterluithTaylorseries butheresketchofheuristicproof
fromgeometry deduce siulxtel siuxcosycosxsiuycoslxtyl
cosxc.su sinxsiuy
settingyeux givesus arecursionrelation
Siu lutilxlsiuxcosuxtcosxsiuuxcoslln.nl
x cosxcosux sinxsinux
solving thisrecursionleadsto de Moine formulas
siuux EI.tn lI1sinkxcosn Ioddcosux E l 11 I sinkcos xE o
keven
set y ux thenu
siny 2 til I sine costh
k oHdd fulfil ysin xzX and cost 1 forsmall fromgeometry
hasc i similar for cosyk o
Kodd
this gives us Eilers formula eit cosxisiuxcosxE.at feint 1 0
six II f Ilk 2K1113 any complexnumber can bewritten as 2 reich r 20 Oeyar
note a few eine so coscetising cosmetisinnydeMoine
say u 2 coscetisine is11
costet2icosesince since
real and imaginarypartshavetobe thesame
cos2q Coste since
3 siu2q 2 cosysince
complexroots of 2 3 2 0 z
write 1 e2rik for 4 0,1 2,3y's niYk.ei EFF fork 3we'rebackat 1
ei ik 2
23 1 has threesolutions z 1 z e
iz e
i
Im2
if RetE
3
Zz ein
ingen e fork O u l are the n complexroots of zh 1similar complex logarithm be2
write 2 reicet2rik keys
cuz In reietuik lnr iq 2rik.ie it hasmultiplevalues
e g butit wife1mi
if 2rik ke
the so called principalvalue of biz is def byusing E veilwith race er
multiplication z er ein z reinIn
ileitez tizz3 Z zz Kree a2
stretchingandrotating zRe
232 z IT z 3E7ei z a file i