Question

Express the complex number `(1+sqrt3i)^2/(sqrt3 -i` in the form of a + ib. Hence, find the modulus and argument of the complex number.

Answer 1

Let, z = `(1+sqrt3i)^2/((sqrt3 -i))`

= `(1+3i^2 + 2sqrt3 i )/((sqrt3- i))`

= `(1- 3 + 2sqrt3 i )/(sqrt3- i)`

= `(-2+2sqrt3 i)/ (sqrt3- i) xx ( sqrt3 + i)/ (sqrt3 + i)`

= `(-2 sqrt3 + 6i - 2i + 2sqrt3 i^2)/(3-i^2)`

= `(-4 sqrt3 + 4i)/(3 + 1)`

=  `(4 (-sqrt3 +i))/(4)`

z = - `sqrt3 + i`

Here,  a = ` - sqrt3` , b = 1

Now, Modulus, | z | = `sqrt( a^2 + b^2)`

= `sqrt(( -sqrt3)^2 + (1)^2)`

= `sqrt(3 + 1) = sqrt(4)`

= 2

Argurement = - `π + tan^-1 |(b)/(a)|`

= - `π + tan^-1 |(1)/-sqrt(3)|`

= `-π + tan^- 1  1/sqrt3`

= -π + `(π)/(6)`

= - `(5π)/(6)`