First we want to convert the factors into trigonometric form.
We can do this using the formula:
#a+bi=r(cos(theta)+isin(theta))#
where #r=sqrt(a^2+b^2)# and #theta=tan^-1(b/a)# and adjusting #theta# to the appropriate quadrant.
#5+7i=r_1(cos(theta_1)+isin(theta_1))#
#r_1=sqrt74~=8.6, theta_1=tan^-1(7/5)~=54.5°#
#9-i=r_2(cos(theta_2)+isin(theta_2))#
#r_2=sqrt(82)~=9.1, theta_2=tan^-1(-1/9)~=-6.3°#
If we have two complex numbers in polar form like this,
#z_1=r_1(cos(phi_1)+isin(phi_1))#
#z_2=r_2(cos(phi_2)+isin(phi_2))#
the product of them will be:
#z_1*z_2=r_1r_2(cos(phi_1+phi_2)+isin(phi_1+phi_2))#
Applying this to our numbers, we get:
#(5+7i)(9-i)=r_1r_2(cos(theta_1+theta_2)+isin(theta_1+theta_2))#
#r_1r_2=sqrt74sqrt82~=77.9, theta_1+theta_2~=54.5°-6.3°~=48.1°#
So, #(5-7i)(9-i)~=77.9(cos(48.1°)+isin(48.1°))#
