Fine the quotient #(z_1)/(z_2)# Thanks?!!
1 Answer
Here is how you can divide two complex numbers in polar form...
Explanation:
Multiplying by a complex number is a combination of rotation and scaling in the complex plane.
Note that:
#(cos alpha + i sin alpha)(cos beta + i sin beta)=e^(ialpha) * e^(ibeta) = e^(i(alpha+beta)) = cos(alpha+beta)+i sin(alpha+beta)#
Similarly:
#(cos alpha + i sin alpha)/(cos beta + i sin beta)=e^(ialpha) / e^(ibeta) = e^(i(alpha-beta)) = cos(alpha-beta)+isin(alpha-beta)#
So to multiply two complex numbers in polar form, we multiply their radii and add their angles, thereby combining scaling and rotation:
#r_1 (cos alpha + i sin alpha) * r_2 (cos beta + i sin beta)#
#=r_1 r_2 (cos(alpha+beta) + i sin(alpha+beta))#
Similarly to divide two complex numbers, we divide their radii and subtract their angles:
#(r_1 (cos alpha+isin alpha))/(r_2 (cos beta+i sin beta)) = r_1/r_2 (cos(alpha-beta) + i sin(alpha-beta))#
So for the given example:
#z_1 = 1/8(cos((2pi)/3)+isin((2pi)/3))#
#z_2 = 1/3(cos(pi/4)+isin(pi/4))#
put:
#{ (r_1 = 1/8), (alpha = (2pi)/3), (r_2 = 1/3), (beta = pi/4) :}#
to find:
#z_1/z_2 = r_1/r_2(cos(alpha-beta)+isin(alpha-beta))#