(2((cos80^0+isin80^0))^3=2^3(cos80^0+isin80^0)^3(2((cos800+isin800))3=23(cos800+isin800)3
By De-Moivre's theorem;
(cosA+isinA)^n=cosnA+isinnA(cosA+isinA)n=cosnA+isinnA
=8(cos(3xx80^0)+isin(3xx80^0))=8(cos(3×800)+isin(3×800))
=8(cos240^0+isin240^0)=8(cos2400+isin2400)
240=180+60
cos240^0=cos(180^0+60^0)cos2400=cos(1800+600)
=-cos60^0=-1/2=−cos600=−12
sin240^0=sin(180^0+60^0)sin2400=sin(1800+600)
=-sin60^0=-sqrt3/2=−sin600=−√32
Thus,
8(cos240^0+isin240^0)=8xx(-1/2+i(-sqrt3)/2)8(cos2400+isin2400)=8×(−12+i−√32)
=-4-4isqrt3=−4−4i√3
(2((cos80^0+isin80^0))^3=-4-4isqrt3(2((cos800+isin800))3=−4−4i√3
