In an amplitude modulator circuit, the carrier wave is given by $C\left(t\right)=4\sin \left(20000\pi t\right)$ while modulating signal is given by, $\mathrm{m}\left(\mathrm{t}\right)=2\sin \left(2000\pi t\right)$. The value of modulation index and lower side band frequency are

Given: carrier wave $c\left(t\right)=4\sin \left(20000\pi t\right)$

Modulating signal, $m\left(t\right)=2\sin \left(2000\pi t\right)$

So, carrier wave's amplitude and frequency are $A_c=4V$

${\omega }_c=20000\pi =2\pi \times {10}^4 rad/s$

$\Rightarrow f_c=\frac{{\omega }_c}{2\pi }={10}^{4}Hz=10kHz$

And modulating signals amplitude and frequency are 

$A_m=2V,w_m=2000\pi =2\pi \times {10}^{3}rad/s$

$\Rightarrow f_m=\frac{{\omega }_m}{2\pi }={10}^{3}Hz=1kHz$

So, modulating Index: $m=\frac{A_m}{A_c}=\frac{2}{4}=\frac{1}{2}=0.5$

lower side and upper side band frequency:

$f_{LSB}=f_c-f_m$ $=10-1=9 kHz$

$f_{USB}=f_c+f_m=10+1=11 kHz$