In amplitude modulation equation of mesenger wave `A_(o)sinomega_(m)t` and carrier wave `A_(C)cosomega_(C)t` the equation of amplitude modulated wave is

Options

A. `x=A_(c)cosomegact+(A_(0))/(2)[sin(omegam+omegac)t+sin(omegac-omegam)t]`
B. `x=A_(c)cosomegact-(A_(0))/(2)[sin(omegam+omegac)t+sin(omegac-omegam)t]`
C. `x=Acsinomegact+(A_(0))/(4)[sin(omegam+omegac)t+sin(omegac-omegam)t]`
D. `x=Acsinomegact+(A_(0))/(4)[sin(omegam+omegac)t+sin(omegac-omegam)t]`

Answer

To find the equation of the amplitude modulated wave given the messenger wave and the carrier wave, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Waves**:
- Messenger wave: \( A_0 \sin(\omega_m t) \)
- Carrier wave: \( A_C \cos(\omega_C t) \)

2. **Formulate the Amplitude Modulated Wave**:
- The amplitude modulated wave (AM wave) can be expressed as:
\[ X(t) = A_C \cos(\omega_C t) + A_0 \sin(\omega_m t) \cos(\omega_C t) \]

3. **Use Trigonometric Identity**:
- We can use the trigonometric identity:
\[ \sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)] \]
- Applying this identity to the term \( A_0 \sin(\omega_m t) \cos(\omega_C t) \):
\[ A_0 \sin(\omega_m t) \cos(\omega_C t) = \frac{A_0}{2} [\sin(\omega_m t + \omega_C t) + \sin(\omega_C t - \omega_m t)] \]

4. **Combine the Terms**:
- Substitute back into the equation for \( X(t) \):
\[ X(t) = A_C \cos(\omega_C t) + \frac{A_0}{2} [\sin(\omega_m t + \omega_C t) + \sin(\omega_C t - \omega_m t)] \]

5. **Final Expression**:
- Thus, the equation of the amplitude modulated wave is:
\[ X(t) = A_C \cos(\omega_C t) + \frac{A_0}{2} \sin(\omega_m + \omega_C)t + \frac{A_0}{2} \sin(\omega_C - \omega_m)t \]

### Final Answer:
The equation of the amplitude modulated wave is:
\[ X(t) = A_C \cos(\omega_C t) + \frac{A_0}{2} \sin(\omega_m + \omega_C)t + \frac{A_0}{2} \sin(\omega_C - \omega_m)t \]