Question

Answer the following:

Find the modulus and argument of a complex number and express it in the polar form.

`(-1 - "i")/sqrt(2)`

Answer 1

Let z = `(-1 - "i")/sqrt(2) = (-1)/sqrt(2) - 1/sqrt(2)"i"`

This is of the form a + bi, where

a = `(-1)/sqrt(2)`, b = `(-1)/sqrt(2)`

∴ modulus = r

= `sqrt("a"^2 + "b"^2)`

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

= `sqrt(1/2 + 1/2)`

= 1

If θ is the amplitude, then

cos θ = `"a"/"r" = -1/sqrt(2)`

and sin θ = `"b"/"r" = -1/sqrt(2)`

`∴ θ = (5pi)/4  ...[(because cos  (5pi)/4 = cos(pi + pi/4) = -cos  pi/4 = -1/sqrt(2)),(and sin  (5pi)/4 = sin (pi + pi/4) = -sin  pi/4 = -1/sqrt(2))]`

∴ the polar form of z = r(cos θ + i sin θ)

= `1(cos  (5pi)/4 + "i" sin  (5pi)/4)`