Question

Find the modulus and argument of a complex number and express it in the polar form.
1 + √3i
_____
     2​

Answer 1

Answer:

1+3i/1-2i

= 1+3i/1-2i × 1+2i/1+2i

= (1+3i)(1+2i) / (1-2i)(1+2i)

= -5+5i / 1^2 - (2i)^2

= -5+5i / 5

= -1+i

modulus =

∴ r \begin{gathered}\sqrt{x^2+y^2}\\ \\=\sqrt{(-1)^2+1^2} \\\\=\sqrt{2}\end{gathered}

x

2

+y

2

=

(−1)

2

+1

2

=

2

Ф = tan^-1 (b/a)

= tan^-1 (1/-1)

= tan^-1 (-1)

∴ Ф = -45°

Polar form is given by,

z = r (cos Ф + i sin Ф)

= √2 [cos (-45° ) + i sin (-45 ° )]

= √2 [cos (45° ) - i sin (45 ° )]

∴ z= √2 e^{-i π/4}