Question

Answer the following:

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

6 − i

Answer 1

Let z = 6 − i

∴ a = 6, b = − 1, a > 0, b < 0

∴ |z| = r

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

= `sqrt(6^2 + (-1)^2)`

= `sqrt(36 + 1)`

= `sqrt(37)`

Here (6, −1) lies in 4^th quadrant

∴ amp (z) = θ = `2pi + tan^-1 ("b"/"a")`

= `2pi + tan^-1 ((-1)/6)`

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

= `sqrt(37)(cos theta + "i" sin theta)`

where θ = `2pi + tan^-1 (-1/6)`