A signal `Acosomegat` is transmitted using `v_(0) sinomega_(0)t` modulated (AM) signal is:

Options

A. `v_(0)sinomega_(0)t+ (A)/(2)sin(omega_(0)-omega)t+ (A)/(2)(omega_(0)+omega)t`
B. `v_(0)sin[omega_(0)(1+0.01Asinomegat)t]`
C. `v_(0)sinomega_(0)t+ Acosomegat`
D. `(v_(0)+ A)cosomegatsinomega_(0)t`

Explanation

To solve the problem of determining the amplitude-modulated (AM) signal when a message 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 Message and Carrier Signals**:
- The message signal is given as \( m(t) = A \cos(\omega t) \).
- The carrier signal is given as \( c(t) = V_0 \sin(\omega_0 t) \).

2. **Understand Amplitude Modulation**:
- In amplitude modulation, the amplitude of the carrier wave is varied in accordance with the message signal. The general form of an AM wave can be expressed as:
\[ s(t) = A_c \left(1 + m(t)\right) c(t) \]
- Here, \( A_c \) is the amplitude of the carrier wave, \( m(t) \) is the normalized message signal, and \( c(t) \) is the carrier wave.

3. **Substituting the Signals**:
- Substitute \( m(t) \) and \( c(t) \) into the AM wave equation:
\[ s(t) = V_0 \left(1 + \frac{A}{V_0} \cos(\omega t)\right) \sin(\omega_0 t) \]

4. **Simplifying the Expression**:
- This can be expanded as:
\[ s(t) = V_0 \sin(\omega_0 t) + A \cos(\omega t) \sin(\omega_0 t) \]

5. **Using Trigonometric Identities**:
- To simplify \( A \cos(\omega t) \sin(\omega_0 t) \), we can use the identity:
\[ \sin A \cos B = \frac{1}{2} \left( \sin(A + B) + \sin(A - B) \right) \]
- Applying this identity:
\[ A \cos(\omega t) \sin(\omega_0 t) = \frac{A}{2} \left( \sin(\omega_0 t + \omega t) - \sin(\omega_0 t - \omega t) \right) \]

6. **Final Expression**:
- Therefore, the complete expression for the AM wave becomes:
\[ s(t) = V_0 \sin(\omega_0 t) + \frac{A}{2} \left( \sin((\omega_0 + \omega)t) - \sin((\omega_0 - \omega)t) \right) \]

7. **Conclusion**:
- The final amplitude-modulated signal is:
\[ s(t) = V_0 \sin(\omega_0 t) + \frac{A}{2} \sin((\omega_0 - \omega)t) - \frac{A}{2} \sin((\omega_0 + \omega)t) \]