Is it possible to add two (different) complex numbers given in trigonometric form directly without changing them in other forms and get a new complex number in trig form? For example: #rho_1[cos(alpha)+isin(alpha)]+rho_2[cos(beta)+isin(beta)]=?# Thanks!

1 Answer
Cesareo R.
Sep 20, 2016

See below.

Explanation:

Of course

#rho_1[cos(alpha)+isin(alpha)]+rho_2[cos(beta)+isin(beta)]=#
#=rho_1cos alpha + rho_2 cos beta + i(rho_1 sin alpha + rho_2 sin beta) =#

#=rho_(12) (cos phi + i sin phi)#

Calling #z_1=rho_1[cos(alpha)+isin(alpha)]# and #z_2=rho_2[cos(beta)+isin(beta)]# we have

#z_1+z_2 = (rho_1 cos alpha + rho_2 cos beta)+i(rho_1 sinalpha+rho_2 sin beta) = z_(12)#.

Now, any complex number can be represented as

#z_(12) = abs(z_(12))e^(i angle z_(12)) = rho_(12)e^(i phi)# where

#rho_(12) = sqrt( (rho_1 cos alpha + rho_2 cos beta)^2+(rho_1 sinalpha+rho_2 sin beta)^2) = sqrt(rho_1^2 + rho_2^2 + 2 rho_1 rho_2 Cos(alpha - beta))#

and

#phi = arctan((rho_1 sinalpha+rho_2 sin beta)/(rho_1 cos alpha + rho_2 cos beta))#

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