A complex wave `y= 3 sin^2t cos500t` is formed by superposition of how many waves?

Answer

To solve the problem of how many waves are superimposed to form the complex wave \( y = 3 \sin^2 t \cos 500t \), we will break down the expression step by step.

### Step-by-Step Solution:

1. **Rewrite the Complex Wave**:
The given wave equation is:
\[ y = 3 \sin^2 t \cos 500t \]
We can use the trigonometric identity for \( \sin^2 t \):
\[ \sin^2 t = \frac{1 - \cos 2t}{2} \]
Substituting this into the equation gives:
\[ y = 3 \left(\frac{1 - \cos 2t}{2}\right) \cos 500t \]

2. **Simplify the Expression**:
Simplifying the equation further:
\[ y = \frac{3}{2} (1 - \cos 2t) \cos 500t \]
This can be expanded as:
\[ y = \frac{3}{2} \cos 500t - \frac{3}{2} \cos 2t \cos 500t \]

3. **Use the Product-to-Sum Formulas**:
To simplify the term \( \cos 2t \cos 500t \), we can use the product-to-sum identities:
\[ \cos A \cos B = \frac{1}{2} (\cos(A + B) + \cos(A - B)) \]
Here, \( A = 2t \) and \( B = 500t \):
\[ \cos 2t \cos 500t = \frac{1}{2} (\cos(502t) + \cos(498t)) \]
Substituting this back into the equation gives:
\[ y = \frac{3}{2} \cos 500t - \frac{3}{4} (\cos(502t) + \cos(498t)) \]

4. **Combine the Terms**:
Now we can rewrite the equation:
\[ y = \frac{3}{2} \cos 500t - \frac{3}{4} \cos 502t - \frac{3}{4} \cos 498t \]

5. **Identify the Individual Waves**:
The equation can now be seen as a superposition of three waves:
- \( \frac{3}{2} \cos 500t \)
- \( -\frac{3}{4} \cos 502t \)
- \( -\frac{3}{4} \cos 498t \)

6. **Count the Waves**:
From the final expression, we can conclude that there are three distinct wave components in the superposition.

### Final Answer:
The complex wave \( y = 3 \sin^2 t \cos 500t \) is formed by the superposition of **3 waves**.

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