Question

Find the square root of `8(cos45^@+isin45^@)` in the form `a+ib`?

Answer 1

`sqrt(8(cos45^@+isin45^@))=sqrt(2sqrt2+2)-isqrt(2sqrt2-2)`

Explanation:

We use here DeMoivre's Theorem according to which

`[r(costheta+isintheta)]^n=r^n(cosntheta+isintheta)`

Hence `sqrt(8(cos45^@+isin45^@))`

= `sqrt8(cos(45^@/2)+isin(45^@/2))`

= `2sqrt2(cos22.5^@+isin22.5^@)`

= `2sqrt2(sqrt(2+sqrt2)/2+i*sqrt(2-sqrt2)/2)`

= `sqrt(2sqrt2+2)-isqrt(2sqrt2-2)`