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