Find the product

Sinx . Sin(x+d) . Sin(x+2d) .Sin(x+3d)...............

To find the product \( \sin x \cdot \sin(x + d) \cdot \sin(x + 2d) \cdot \sin(x + 3d) \cdots \), we can utilize some properties of trigonometric functions and a bit of algebraic manipulation. This product can be expressed in a more manageable form using the sine addition formula and some identities. Let's break it down step by step.

Understanding the Product of Sine Functions

The expression involves multiple sine functions, each offset by a constant \( d \). The general term in this product can be represented as \( \sin(x + nd) \) where \( n \) takes on integer values starting from 0. The product can be expressed as:

• First term: \( \sin x \)
• Second term: \( \sin(x + d) \)
• Third term: \( \sin(x + 2d) \)
• Fourth term: \( \sin(x + 3d) \)
• And so on...

Using the Product-to-Sum Formulas

To simplify the product, we can use the product-to-sum identities, which relate products of sine functions to sums. However, for a product of sine functions with a constant difference, we can also use a specific formula:

For \( n \) terms, the product can be expressed as:

Product = \frac{1}{2^{n-1}} \cdot \sin\left(\frac{nd}{2}\right) \cdot \sin\left(x + \frac{(n-1)d}{2}\right)

Deriving the Formula

Let’s derive this formula for a clearer understanding. Consider the case where we have \( n \) terms:

1. The product can be rewritten using the sine addition formula:

\( \sin A \cdot \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)] \)

2. By applying this iteratively, we can reduce the product step by step.

3. After simplification, we arrive at the general form mentioned earlier.

Example Calculation

Let’s say we want to calculate the product for \( n = 4 \) and \( d = \frac{\pi}{4} \):

We have:

• First term: \( \sin x \)
• Second term: \( \sin\left(x + \frac{\pi}{4}\right) \)
• Third term: \( \sin\left(x + \frac{\pi}{2}\right) \)
• Fourth term: \( \sin\left(x + \frac{3\pi}{4}\right) \)

Using the derived formula, we can calculate:

Product = \( \frac{1}{2^{4-1}} \cdot \sin(4 \cdot \frac{\pi}{8}) \cdot \sin\left(x + \frac{3\pi}{8}\right) \)

Which simplifies to:

Product = \( \frac{1}{8} \cdot \sin\left(\frac{\pi}{2}\right) \cdot \sin\left(x + \frac{3\pi}{8}\right) \)

Since \( \sin\left(\frac{\pi}{2}\right) = 1 \), the product simplifies further.

Final Thoughts

In summary, the product of sine functions with a constant difference can be expressed in a more manageable form using trigonometric identities. This approach not only simplifies calculations but also provides deeper insights into the behavior of sine functions in periodic contexts. If you have any further questions or need clarification on specific steps, feel free to ask!