Answer to Question #117716 in Complex Analysis for Gloria Ampofowaa

Question #117716
Find the modulus and the principal argument of each of the given complex numbers.
(a) 3−4i, (b) −2+i, (c) 1√ , (d) 7−i , 1+i 3 −4−3i
(e) 5(cos π/3 + i sin π/3), (f) cos 2π/3 − sin 2π
1
Expert's answer
2020-05-25T18:48:07-0400

a) z=34iz=3-4i

z=32+42=5|z|=\sqrt{3^2+4^2}=5

cosθ=3/5,sinθ=4/5cos\theta=3/5, sin\theta=-4/5

θ=53°\theta=-53\degree


b) z=2+iz=-2+i

z=22+1=5|z|=\sqrt{2^2+1}=\sqrt{5}

cosθ=2/5,sinθ=1/5cos\theta=-2/\sqrt{5}, sin\theta=1/\sqrt{5}

θ=153°\theta=153\degree


d) z=7iz=7-i

z=72+1=52|z|=\sqrt{7^2+1}=5\sqrt{2}

cosθ=752,sinθ=152cos\theta=\frac {7}{5\sqrt{2}}, sin\theta=-\frac {1}{5\sqrt{2}}

θ=1°\theta=-1\degree


z=1+iz=1+i

z=1+1=2|z|=\sqrt{1+1}=\sqrt{2}

cosθ=sinθ=1/2cos\theta=sin\theta=1/\sqrt{2}

θ=45°\theta=45\degree


z=43iz=-4-3i

z=42+32=5|z|=\sqrt{4^2+3^2}=5

cosθ=4/5,sinθ=3/5cos\theta=-4/5, sin\theta=-3/5

θ=143°\theta=-143\degree


e) z=5(cosπ/3+isinπ/3)z=5(cos π/3 + i sin π/3)

z=5cos2π/3+sin2π/3=5|z|=5\sqrt{cos^2\pi/3+sin^2\pi/3}=5

θ=π/3\theta=\pi/3


f) z=cos2π/3isin2π/3z=cos 2π/3 − isin 2π/3

z=cos22π/3+sin22π/3=1|z|=\sqrt{cos^22\pi/3+sin^22\pi/3}=1

θ=2π/3\theta=-2\pi/3


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