Question

What does the equation `z_2/z_1=r_2/r_1(cos(θ_2-θ_1)+isin(θ_2-θ_1))` represent?

Is it correct to say this represents the division of polar/trig forms?

Answer 1

Yes, it is correct to say this represents the division of polar/trig forms.

Explanation:

`z_2/z_1=r_2/r_1(cos(theta_2-theta_1)+isin(theta_2-theta_1))`

represents division of polar/trig forms of two polar numbers

`z_2=r_2(costheta_2+isintheta_2)` and `z_1=r_1(costheta_1+isintheta_1)`

Observe that `z_2/z_1=(r_2(costheta_2+isintheta_2))/(r_1(costheta_1+isintheta_1))`

Now rationalizing the denominator by multiplying by cojugate of denominator, we get

`r_2/r_1((costheta_2+isintheta_2)(costheta_1-isintheta_1))/((costheta_1+isintheta_1)(costheta_1-isintheta_1))`

= `r_2/r_1(costheta_2costheta_1+isintheta_2costheta_1-icostheta_2sintheta_1-i^2sintheta_2sintheta_1)/(cos^2theta_1+sin^2theta_1)`

= `r_2/r_1((costheta_2costheta_1+sintheta_2sintheta_1)+i(sintheta_2costheta_1-icostheta_2sintheta_1))/(cos^2theta_1+sin^2theta_1)`

= `r_2/r_1(cos(theta_2-theta_1)+isin(theta_2-theta_1))`