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 multipication leads us to
{r1⋅r2}{(cosα+isinα)⋅(cosβ+isinβ)} or
{r1⋅r2}{(cosαcosβ+i2sinαsinβ)+i(cosαsinβ+cosβsinα)) or
{r1⋅r2}{(cosαcosβ−sinαsinβ)+i(cosαsinβ+cosβsinα)) or
(r1⋅r2)⋅(cos(α+β)+isin(α+β)) or
z1⋅z2 is given by (r1⋅r2,(α+β))
So for multiplication of complex number z1 and z2 , take new angle as (α+β) and modulus os r1⋅r2 of the modulus of two numbers.
Here 7−3i can be written as r1(cosα+isinα) where r1=√72+(−3)2=√58 and α=tan−1(−37)
and 5−i can be written as r2(cosβ+isinβ) where r2=√52+(−1)2=√26 and β=tan−1(−15)
and z1⋅z2=√58⋅(√26)(cosθ+isinθ), where θ=α+β
Hence, tanθ=tan(α+β)=tanα+tanβ1−tanαtanβ=−37+(−15)1−(−37×(−15))=−22353235=−2232=−1116.
Hence, (7−3i)(5−i)=√58×26(cosθ+isinθ)
= 2√377(cosθ+isinθ), where θ=tan−1(−1116)
