I have a triangle $ABC$ in a complex plane. The arrangement of vertices is in a counterclockwise direction. The coordinates of $A$,$B$,$C$ are $z_A$,$z_B$,$z_C$ respectively. It is given that length of side $AB = c, AC = b$ and $\angle BAC = \alpha$. I need to find the other sides and angles of the triangle. How do i do this with complex numbers?
I know with trigonometry using the sine and cosine rules, all the length and angles can be derived. but how to get it using complex numbers?
Here is the connection with the law of cosines. You can see which quantities correspond to which.
You have $z_B-z_A=ce^{i\alpha_1}$ and $z_C-z_A=be^{i\alpha_2}$, hence $z_A-z_C=be^{i(\alpha_2+\pi)}$. Clearly $\alpha_2-\alpha_1=\alpha$. Then $z_B-z_C=(z_B-z_A)+(z_A-z_C)$ and $$ a^2=|z_B-z_C|^2=(z_B-z_C)\overline{(z_B-z_C)}=(z_B-z_A)\overline{(z_B-z_A)}+(z_A-z_C)\overline{(z_A-z_C)}+2Re[(z_B-z_A)\overline{(z_A-z_C)}]= $$ $$ =c^2+b^2+2bcRe(e^{i(\alpha_1-\alpha_2-\pi)})=c^2+b^2-2bc\cos\alpha $$