Answer to Question #301381 in Complex Analysis for pianowoman

Question #301381

1a) Find the real numbers a and b such that

i5(3 - 2i)2 - (a - bi) = a - bi.


b. Let z1 =  √3 + i and z2 = √3 - i. Show that

z161 + z261 = 261√3.


(Remember that: cos(- θ) = cos(θ), sin(θ) = -sin(θ) = -sin(θ), cos(2kπ + θ) = cos(θ) and sin(2kπ + θ) = sin(θ) )

1
Expert's answer
2022-02-23T13:06:06-0500

1a)


i5(32i)2(abi)=abi.i^5(3 - 2i)^2 - (a - bi) = a - bi.

i(912i4)=2a2bii(9-12i-4)=2a-2bi

12+5i=2a2bi12+5i=2a-2bi

a=6,b=5/2a=6, b=-5/2

b.


z1=3+i=2(cos(π/6)+isin(π/6))z_1=\sqrt{3}+i=2(\cos (\pi/6)+i\sin (\pi/6))

z2=3i=2(cos(π/6)+isin(π/6))z_2=\sqrt{3}-i=2(\cos (-\pi/6)+i\sin (-\pi/6))

z161=(2(cos(π/6)+isin(π/6)))61z_1^{61}=(2(\cos (\pi/6)+i\sin (\pi/6)))^{61}

=261(cos(61π/6)+isin(61π/6))=2^{61}(\cos (61\pi/6)+i\sin (61\pi/6))

=261(cos(π/6)+isin(π/6))=2^{61}(\cos (\pi/6)+i\sin (\pi/6))

z261=(2(cos(π/6)+isin(π/6)))61z_2^{61}=(2(\cos (-\pi/6)+i\sin (-\pi/6)))^{61}

=261(cos(61π/6)+isin(61π/6))=2^{61}(\cos (-61\pi/6)+i\sin (-61\pi/6))

=261(cos(π/6)+isin(π/6))=2^{61}(\cos (-\pi/6)+i\sin (-\pi/6))

=261(cos(π/6)isin(π/6))=2^{61}(\cos (\pi/6)-i\sin (\pi/6))

z161+z261=261(cos(π/6)+isin(π/6))z_1^{61}+z_2^{61}=2^{61}(\cos (\pi/6)+i\sin (\pi/6))

+261(cos(π/6)isin(π/6))+2^{61}(\cos (\pi/6)-i\sin (\pi/6))

=261(2cos(π/6))=2613=2^{61}(2\cos (\pi/6))=2^{61}\sqrt{3}


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