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

Answer

To determine how many waves are superimposed to form the complex wave \( y = 3 \sin^2 t \cos 500t \), we can follow these steps:

### Step 1: Rewrite the Complex Wave
We start with the given wave equation:
\[ y = 3 \sin^2 t \cos 500t \]

### Step 2: Use Trigonometric Identities
We can use the trigonometric identity:
\[ \sin^2 t = \frac{1 - \cos(2t)}{2} \]
to rewrite \( \sin^2 t \):
\[ y = 3 \left(\frac{1 - \cos(2t)}{2}\right) \cos 500t \]
This simplifies to:
\[ y = \frac{3}{2} (1 - \cos(2t)) \cos 500t \]

### Step 3: Expand the Expression
Now, we can distribute \( \cos 500t \):
\[ y = \frac{3}{2} \cos 500t - \frac{3}{2} \cos(2t) \cos(500t) \]

### Step 4: Apply Another Trigonometric Identity
Next, we can use the product-to-sum identities for the term \( \cos(2t) \cos(500t) \):
\[ \cos A \cos B = \frac{1}{2} (\cos(A+B) + \cos(A-B)) \]
Applying this identity:
\[ \cos(2t) \cos(500t) = \frac{1}{2} (\cos(502t) + \cos(498t)) \]
Thus, we can rewrite \( y \):
\[ y = \frac{3}{2} \cos 500t - \frac{3}{4} (\cos(502t) + \cos(498t)) \]

### Step 5: Combine the Terms
Now, we can express \( y \) as:
\[ y = \frac{3}{2} \cos 500t - \frac{3}{4} \cos(502t) - \frac{3}{4} \cos(498t) \]

### Step 6: Identify the Number of Waves
In the final expression, we can see that we have three distinct cosine terms:
1. \( \cos 500t \)
2. \( \cos 502t \)
3. \( \cos 498t \)

Thus, the complex wave \( y = 3 \sin^2 t \cos 500t \) is formed by the superposition of **three waves**.

### Final Answer
The number of waves superimposed to form the complex wave is **3**.

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