A signal Acos `omega`t is transmitted using `upsilon_(0)sin omega_(0)t` as carrier wave. The correct amplitude modulated (AM) signal is :

Options

A. `upsilon_(0)sin omega_(0)t+(A)/(2)sin(omega_(0)-omega)t+(A)/(2)sin(omega_(0)+omega)t`
B. `upsilon_(0)sin[omega_(0)(1+0.01A sin omega t)t]`
C. `upsilon_(0)sin omega_(0)t+A cos omega t`
D. `(upsilon_(0)+A)cos omega t sin omega_(0)t`

Answer

To solve the problem of finding the amplitude modulated (AM) signal when a signal \( A \cos(\omega t) \) is transmitted using a carrier wave \( v_0 \sin(\omega_0 t) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Signals**:
- The message signal is \( A \cos(\omega t) \).
- The carrier wave is \( v_0 \sin(\omega_0 t) \).

2. **Understand Amplitude Modulation**:
- In amplitude modulation, the message signal modulates the amplitude of the carrier wave. The modulated signal can be expressed as:
\[ \text{AM Signal} = (v_0 + A \cos(\omega t)) \sin(\omega_0 t) \]

3. **Expand the Expression**:
- We can distribute the sine function:
\[ \text{AM Signal} = v_0 \sin(\omega_0 t) + A \cos(\omega t) \sin(\omega_0 t) \]

4. **Use Trigonometric Identities**:
- To simplify \( A \cos(\omega t) \sin(\omega_0 t) \), we can use the product-to-sum identities:
\[ A \cos(\omega t) \sin(\omega_0 t) = \frac{A}{2} \left( \sin(\omega_0 t + \omega t) - \sin(\omega_0 t - \omega t) \right) \]
- Therefore, the modulated signal becomes:
\[ \text{AM Signal} = v_0 \sin(\omega_0 t) + \frac{A}{2} \left( \sin((\omega_0 + \omega)t) - \sin((\omega_0 - \omega)t) \right) \]

5. **Final Expression**:
- The final expression for the amplitude modulated signal is:
\[ \text{AM Signal} = v_0 \sin(\omega_0 t) + \frac{A}{2} \sin((\omega_0 + \omega)t) - \frac{A}{2} \sin((\omega_0 - \omega)t) \]