Question

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

Answer 1

a) $z=3-4i$

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

$cos\theta=3/5, sin\theta=-4/5$

$\theta=-53\degree$

b) $z=-2+i$

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

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

$\theta=153\degree$

d) $z=7-i$

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

$cos\theta=\frac {7}{5\sqrt{2}}, sin\theta=-\frac {1}{5\sqrt{2}}$

$\theta=-1\degree$

$z=1+i$

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

$cos\theta=sin\theta=1/\sqrt{2}$

$\theta=45\degree$

$z=-4-3i$

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

$cos\theta=-4/5, sin\theta=-3/5$

$\theta=-143\degree$

e) $z=5(cos π/3 + i sin π/3)$

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

$\theta=\pi/3$

f) $z=cos 2π/3 − isin 2π/3$

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

$\theta=-2\pi/3$