Given the electric field of a complete amplitude modulated wave as
` vecE= hati E_C ( 1 + (E_m)/(E_C) cos omega_m t ) cos omega_c t`.
Where the subscript c stands for the carrier wave and m for the modulating signal . The frequencies present in the modulated wave are

Options

A. `omega_c and sqrt( omega_c^(2) + omega_m^(2))`
B. `omega_c , omega_c + omega_m and omega_c - omega_m`
C. `omega_c and omega_m `
D. `omega_c and sqrt( omega_c omega_m)`

Explanation

To find the frequencies present in the given amplitude modulated wave, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Electric Field Equation**:
The electric field of the amplitude modulated wave is given by:
\[ \vec{E} = \hat{i} E_C \left(1 + \frac{E_m}{E_C} \cos(\omega_m t)\right) \cos(\omega_c t) \]
Here, \(E_C\) is the amplitude of the carrier wave, \(E_m\) is the amplitude of the modulating signal, \(\omega_c\) is the angular frequency of the carrier wave, and \(\omega_m\) is the angular frequency of the modulating signal.

2. **Identify the Frequencies**:
In amplitude modulation, the modulated wave consists of:
- The **carrier frequency**: This is represented by \(\omega_c\).
- The **upper sideband frequency**: This is given by \(\omega_c + \omega_m\).
- The **lower sideband frequency**: This is given by \(\omega_c - \omega_m\).

3. **List the Frequencies Present**:
Therefore, the frequencies present in the modulated wave are:
- Carrier Frequency: \(\omega_c\)
- Upper Sideband Frequency: \(\omega_c + \omega_m\)
- Lower Sideband Frequency: \(\omega_c - \omega_m\)

4. **Conclusion**:
The three frequencies present in the amplitude modulated wave are:
\[ \omega_c, \quad \omega_c + \omega_m, \quad \omega_c - \omega_m \]

### Final Answer:
The frequencies present in the modulated wave are:
- \(\omega_c\)
- \(\omega_c + \omega_m\)
- \(\omega_c - \omega_m\)