Question

Answer the following:

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

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

Answer 1

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

∴ a = `(1)/sqrt(2)`, b = `(1)/sqrt(2)`, a > 0, b > 0

∴ |z| = r

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

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

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

= 1

Here `(1/sqrt(2), 1/sqrt(2))` lies in 1^st quadrant

amp (z) = θ = `tan^-1("b"/"a")`

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

= tan^–1(1) = `pi/4`

∴ θ = 45° = `pi/4`

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

= 1(cos 45° + i sin 45°)

= `1(cos  pi/4 + "i"sin  pi/4)`