9.1 The Moivre’s Theorem
z=r(cosθ+isinθ)zn=rn(cosnθ+isinnθ),n∈N
9.2 Let r=1
z=cosθ+isinθz5=cos5θ+isin5θ=(cosθ+isinθ)5==cos5θ+5icos4θsinθ+10i2cos3θsin2θ++10i3cos2θsin3θ+5i4cosθsin4θ+i5sin5θ==cos5θ−10cos3θsin2θ+5cosθsin4θ++i(5cos4θsinθ−10cos2θsin3θ+sin5θ)cos5θ=cos5θ−10cos3θsin2θ+5cosθsin4θ
z4=cos4θ+isin4θ=(cosθ+isinθ)4==cos4θ+4icos3θsinθ+6i2cos2θsin2θ++6i3cosθsin3θ+i4sin4θ==cos4θ−6cos2θsin2θ+sin4θ++i(4cos3θsinθ−6cosθsin3θ)sin4θ=4cos3θsinθ−6cosθsin3θ
i2=−1,i3=−i,i4=1,i5=i
9.3 a)
z=6w,w<0w=ρ(cosθ+isinθ)z=6ρ(cos6θ+2πk+isin6θ+2πk),k=0,1,2,3,4,5
b)
6−729=6−1⋅36=36−1−1=cosπ+isinπ6−1=cos6π+2πk+sin6π+2πk,k=0,1,2,3,4,56−729=3(cos6π+2πk+sin6π+2πk),k=0,1,2,3,4,5
6−729=3(cos6π+sin6π)=3(23+i21),6−729=3(cos63π+sin63π)=3i,6−729=3(cos65π+sin65π)=3(−23+i21),6−729=3(cos67π+sin67π)=3(−23−i21),6−729=3(cos69π+sin69π)=−3i,6−729=3(cos611π+sin611π)=3(23−i21)
c)
z=cosθ+isinθu+iv=(1+z)(1+z2)==((1+cosθ)+isinθ)⋅(1+cos2θ+2icosθsinθ−sin2θ)==((1+cosθ)+isinθ)⋅((1+cos2θ−sin2θ)+2icosθsinθ)==((1+cosθ)+isinθ)⋅(2cos2θ+2icosθsinθ)==2cos2θ(1+cosθ)−2cosθsin2θ++i(2cosθsinθ(1+cosθ)+2sinθcos2θ)u=2cos2θ(1+cosθ)−2cosθsin2θ==2cosθ(cosθ+cos2θ)v=2cosθsinθ(1+cosθ)+2sinθcos2θ==2sinθcosθ(1+2cosθ)=2cosθ(sinθ+sin2θ)utan23θ=2cosθ(cosθ+cos2θ)cos23θsin23θ==cos23θ2cosθ2sin2θ+sin25θ−sin2θ+sin27θ==cos23θ2cosθ22sin3θcos2θ==2cosθ(2sin23θcos2θ)==2cosθ(sinθ+sin2θ)=v
u2+v2=u=4cos2θ(cosθ+cos2θ)2++4cos2θ(sinθ+sin2θ)2==4cos2θ(cos2θ+2cosθcos2θ+cos22θ++sin2θ+2sinθsin2θ+sin22θ)==4cos2θ(2+2cosθ)=8cos2θ(1+cosθ)==16cos2θcos22θ
Comments