A message signal is superimposed with a carrier signal. The resulting modulated signal c_m(t) is given by ${\mathrm{c}}_{\mathrm{m}}\left(\mathrm{t}\right)={\mathrm{A}}_1\sin \left({\mathrm{\omega }}_1\mathrm{t}\right)+{\mathrm{A}}_2\sin \left({\mathrm{\omega }}_2\mathrm{t}\right)-{\mathrm{A}}_2\sin \left({\mathrm{\omega }}_3\mathrm{t}\right);$ where $\omega$₂ < $\omega$₁ < $\omega$₃. The modulation index and the angular frequency of the message signal respectively are

Given modulating signal
${\mathrm{C}}_{\mathrm{m}}\left(\mathrm{t}\right)={\mathrm{A}}_1\sin \left({\mathrm{w}}_1\mathrm{t}\right)+{\mathrm{A}}_2\sin \left({\mathrm{w}}_2\mathrm{t}\right)-{\mathrm{A}}_3\sin \left({\mathrm{w}}_3\mathrm{t}\right)\to \left(1\right){\mathrm{w}}_1>{\mathrm{w}}_2;{\mathrm{w}}_3>{\mathrm{w}}_1$
We know modulated signal is 
${\mathrm{C}}_{\mathrm{m}}\left(\mathrm{t}\right)={\mathrm{A}}_{\mathrm{c}}\sin \left({\mathrm{w}}_{\mathrm{c}}\mathrm{t}\right)+\frac{{\mathrm{\mu A}}_{\mathrm{c}}}{2}\cos \left({\mathrm{w}}_{\mathrm{c}}-{\mathrm{w}}_{\mathrm{m}}\right)\mathrm{t}-\frac{{\mathrm{\mu A}}_{\mathrm{c}}}{2}\cos \left({\mathrm{w}}_{\mathrm{c}}+{\mathrm{w}}_{\mathrm{m}}\right)\mathrm{t}\to \left(2\right)$
e.q (1) can also be written as;
${\mathrm{C}}_{\mathrm{m}}\left(\mathrm{t}\right)={\mathrm{A}}_1\sin \left({\mathrm{w}}_1\mathrm{t}\right)+{\mathrm{A}}_2\cos \left(\frac{\mathrm{\pi }}{2}+{\mathrm{w}}_2\mathrm{t}\right)-{\mathrm{A}}_3\cos \left(\frac{\mathrm{\pi }}{2}+{\mathrm{w}}_3\mathrm{t}\right)\to \left(3\right)$
From (2) & (3)
$\begin{array}{l}{\mathrm{A}}_1={\mathrm{A}}_{\mathrm{c}};{\mathrm{w}}_{\mathrm{e}}={\mathrm{w}}_1;{\mathrm{w}}_{\mathrm{e}}-{\mathrm{w}}_{\mathrm{m}}={\mathrm{w}}_2\to \left(4\right)\\ \mathrm{\&}{\mathrm{w}}_{\mathrm{e}}+{\mathrm{w}}_{\mathrm{m}}={\mathrm{w}}_3\to \left(5\right)\\ \text{Also }\frac{{\mathrm{\mu A}}_{\mathrm{c}}}{2}={\mathrm{A}}_2\Rightarrow \mathrm{\mu }=\frac{2{\mathrm{A}}_2}{{\mathrm{A}}_{\mathrm{c}}}=\frac{2{\mathrm{A}}_2}{{\mathrm{A}}_1}\to \left(6\right)\end{array}$
eq (5) – eq (4)
$\begin{array}{l}{\mathrm{w}}_3-{\mathrm{w}}_2=\left({\mathrm{w}}_{\mathrm{c}}+{\mathrm{w}}_{\mathrm{m}}\right)-\left({\mathrm{w}}_{\mathrm{c}}-{\mathrm{w}}_{\mathrm{m}}\right)\\ 2{\mathrm{w}}_{\mathrm{m}}={\mathrm{w}}_3-{\mathrm{w}}_2\\ {\mathrm{w}}_{\mathrm{m}}=\frac{{\mathrm{w}}_3-{\mathrm{w}}_2}{2}\end{array}$