(A): If two waves of same amplitude, produce a resultant wave of same amplitude, then the phase difference between them will be `120^(@)`
(R): The resultant amplitude of two waves is equal to sum of amplitude of two waves

Options

A. Both 'A' and 'R' are true and 'R' is the correct, explanation of 'A'.
B. Both 'A' and 'R' are true and 'R' is not the correct explanation of 'A'
C. A' is true and 'R' is false
D. Both 'A' and 'R' are false

Answer

To solve the question, we need to analyze both the assertion (A) and the reason (R) provided.

**Assertion (A):** If two waves of the same amplitude produce a resultant wave of the same amplitude, then the phase difference between them will be \(120^\circ\).

**Reason (R):** The resultant amplitude of two waves is equal to the sum of the amplitudes of the two waves.

### Step-by-Step Solution:

1. **Understanding the Waves:**
- Let the two waves be represented as:
- \( y_1 = a \sin(\omega t) \)
- \( y_2 = a \sin(\omega t + \phi) \)
- Here, \( a \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase difference.

2. **Finding the Resultant Amplitude:**
- The resultant wave \( y \) can be expressed as:
\[ y = y_1 + y_2 = a \sin(\omega t) + a \sin(\omega t + \phi) \]
- Using the sine addition formula, we can rewrite this as:
\[ y = 2a \cos\left(\frac{\phi}{2}\right) \sin\left(\omega t + \frac{\phi}{2}\right) \]
- The resultant amplitude \( A_r \) is given by:
\[ A_r = 2a \cos\left(\frac{\phi}{2}\right) \]

3. **Setting Up the Condition:**
- According to the assertion, the resultant amplitude is equal to the original amplitude \( a \):
\[ A_r = a \]
- Therefore, we have:
\[ 2a \cos\left(\frac{\phi}{2}\right) = a \]
- Dividing both sides by \( a \) (assuming \( a \neq 0 \)):
\[ 2 \cos\left(\frac{\phi}{2}\right) = 1 \]

4. **Solving for Phase Difference:**
- Rearranging gives:
\[ \cos\left(\frac{\phi}{2}\right) = \frac{1}{2} \]
- The angle whose cosine is \( \frac{1}{2} \) is \( 60^\circ \):
\[ \frac{\phi}{2} = 60^\circ \implies \phi = 120^\circ \]
- Thus, the assertion (A) is **true**.

5. **Evaluating the Reason (R):**
- The reason states that the resultant amplitude of two waves is equal to the sum of the amplitudes of the two waves.
- However, from our earlier analysis, we found that:
\[ A_r = 2a \cos\left(\frac{\phi}{2}\right) \neq a + a \]
- Therefore, the reason (R) is **false**.

### Conclusion:
- The assertion (A) is true, while the reason (R) is false. Thus, the correct option is **3: A is true and R is false**.