Answer to Question #117386 in Complex Analysis for Amoah Henry

Question #117386
. Find the modulus and the principal argument of each of the given complex numbers. (a) 3 − 4i, (b) −2 + i, (c)
1 1 + i √ , (d) 3 7 − i −4 − 3i (e) 5(cos π/3 + i sin π/3), (f) cos 2π/3 − sin 2π/3
1
Expert's answer
2020-06-01T19:34:05-0400

(a) 3 − 4i


z=34iz=3-4iz=(3)2+(4)2=5|z|=\sqrt{(3)^2+(-4)^2}=5Arg(z)=arctan(43)=arctan(43)Arg(z)=\arctan(-{4\over 3})=-\arctan({4\over 3})


(b) −2 + i


z=2+iz=-2+iz=(2)2+(1)2=5|z|=\sqrt{(-2)^2+(1)^2}=\sqrt{5}Arg(z)=πarctan(12)Arg(z)=\pi-\arctan({1\over 2})


(c) 1/(1 + i√3)


z=11+i3=1i34z={1\over 1+i\sqrt{3}}={1-i\sqrt{3}\over 4}z=(14)2+(34)2=12|z|=\sqrt{({1\over 4})^2+(-{\sqrt{3}\over 4})^2}={1\over 2}Arg(z)=π3Arg(z)=-{\pi\over 3}


(d) (7 − i)/(−4 − 3i)


z=7i43i=125(7i)(4+3i)=1+iz={7-i\over -4-3i}={1\over 25}(7-i)(-4+3i)=-1+iz=(1)2+(1)2=2|z|=\sqrt{(-1)^2+(1)^2}=\sqrt{2}Arg(z)=ππ4=3π4Arg(z)=\pi-{\pi\over 4}={3\pi\over 4}


(e) 5(cos π/3 + isin π/3)


z=5(cos(π3)+isin(π3))z=5(\cos({\pi\over 3})+i\sin({\pi\over 3}))z=5|z|=5Arg(z)=π3Arg(z)={\pi\over 3}


(f) cos 2π/3 −sin 2π/3.


z=cos(2π3)sin(2π3)=1232z=\cos({2\pi\over 3})-\sin({2\pi\over 3})=-{1\over 2}-{\sqrt{3}\over 2}z=1+32|z|={1+\sqrt{3}\over 2}Arg(z)=πArg(z)=\pi


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