Let us write the two complex numbers in polar coordinates and let them be
z1=r1(cosα+isinα) and z2=r2(cosβ+isinβ)
Here, if two complex numbers are a1+ib1 and a2+ib2 r1=√a21+b21, r2=√a22+b22 and α=tan−1(b1a1), β=tan−1(b2a2)
Their division leads us to
{r1r2}{cosα+isinαcosβ+isinβ} or
{r1r2}{cosα+isinαcosβ+isinβ×cosβ−isinβcosβ−isinβ}
(r1r2)(cosαcosβ+sinαsinβ)+i(sinαcosβ−cosαsinβ)(cos2β+sin2β) or
(r1r2)⋅(cos(α−β)+isin(α−β)) or
z1z2 is given by (r1r2,(α−β))
So for division complex number z1 by z2 , take new angle as (α−β) and modulus the ratio r1r2 of the modulus of two numbers.
Here 2i−7 or −7+2i can be written as r1(cosα+isinα) where r1=√(−7)2+22=√53 and α=tan−1(−27)
and −5i+8 or 8−5i can be written as r2(cosβ+isinβ) where r2=√52+82=√89 and β=tan−1(−58)
and z1z2=√53√89(cosθ+isinθ), where θ=α−β
Hence, tanθ=tan(α−β)=tanα−tanβ1+tanαtanβ=(−27)−(−58)1+(−27)×(−58)=19566656=−1966.
Hence, 2i−7−5i+8=√53√89(cosθ+isinθ)
= 0.7717(cosθ+isinθ), where θ=tan−1(1966)
