Rewriting Products As Sums Or Differences Transforming 20 Sin(20x) Sin(11x)

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Introduction

In the realm of trigonometry, it is often necessary to transform trigonometric expressions from one form to another. One common task is to rewrite a product of trigonometric functions as a sum or difference, or vice versa. This technique is particularly useful in simplifying complex expressions, solving trigonometric equations, and performing integration. In this article, we will delve into the product-to-sum identities and demonstrate how to rewrite the given expression, $20 \sin(20x) \sin(11x)$, as a sum or difference.

Rewriting trigonometric expressions is a fundamental skill in various fields such as physics, engineering, and mathematics. For example, in signal processing, product-to-sum identities are used to analyze and manipulate signals that involve products of sinusoidal functions. Similarly, in quantum mechanics, these identities are useful in simplifying expressions involving wave functions. The ability to transform trigonometric expressions allows us to gain deeper insights into the underlying phenomena and solve complex problems more efficiently. This article aims to provide a comprehensive guide on how to rewrite products of sine and cosine functions into sums or differences, focusing on practical applications and step-by-step explanations.

Product-to-Sum Identities

The product-to-sum identities are a set of trigonometric identities that allow us to express products of trigonometric functions as sums or differences. These identities are derived from the sum and difference formulas for sine and cosine. There are four main product-to-sum identities, which are:

1. $\sin(A) \cos(B) = \frac{1}{2} [\sin(A + B) + \sin(A - B)]$
2. $\cos(A) \sin(B) = \frac{1}{2} [\sin(A + B) - \sin(A - B)]$
3. $\cos(A) \cos(B) = \frac{1}{2} [\cos(A + B) + \cos(A - B)]$
4. $\sin(A) \sin(B) = -\frac{1}{2} [\cos(A + B) - \cos(A - B)]$

These identities are derived using the sum and difference formulas for sine and cosine. Let's briefly review these sum and difference formulas, as they are the foundation for the product-to-sum identities. The sum and difference formulas are:

• $\sin(A + B) = \sin(A) \cos(B) + \cos(A) \sin(B)$
• $\sin(A - B) = \sin(A) \cos(B) - \cos(A) \sin(B)$
• $\cos(A + B) = \cos(A) \cos(B) - \sin(A) \sin(B)$
• $\cos(A - B) = \cos(A) \cos(B) + \sin(A) \sin(B)$

By adding or subtracting these formulas in pairs, we can derive the product-to-sum identities. For example, adding the formulas for $\cos(A + B)$ and $\cos(A - B)$ gives:

$\cos(A + B) + \cos(A - B) = 2 \cos(A) \cos(B)$

Dividing both sides by 2, we obtain the third product-to-sum identity:

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

Similarly, subtracting the formula for $\cos(A + B)$ from $\cos(A - B)$ gives:

$\cos(A - B) - \cos(A + B) = 2 \sin(A) \sin(B)$

Dividing both sides by 2 and multiplying by -1, we get the fourth product-to-sum identity:

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

These identities are essential tools for manipulating trigonometric expressions and are widely used in various applications. Understanding their derivation and application is crucial for mastering trigonometric transformations.

Applying the Product-to-Sum Identity

To rewrite the expression $20 \sin(20x) \sin(11x)$ as a sum or difference, we need to identify which product-to-sum identity is applicable. In this case, we have a product of two sine functions, so we will use the fourth identity:

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

Here, we can identify $A = 20x$ and $B = 11x$. Substituting these values into the identity, we get:

$\sin(20x) \sin(11x) = -\frac{1}{2} [\cos(20x + 11x) - \cos(20x - 11x)]$

Now, we simplify the expression inside the brackets:

$\sin(20x) \sin(11x) = -\frac{1}{2} [\cos(31x) - \cos(9x)]$

We are given the expression $20 \sin(20x) \sin(11x)$, so we multiply both sides of the equation by 20:

$20 \sin(20x) \sin(11x) = 20 \times -\frac{1}{2} [\cos(31x) - \cos(9x)]$

This simplifies to:

$20 \sin(20x) \sin(11x) = -10 [\cos(31x) - \cos(9x)]$

Finally, we distribute the -10 to both terms inside the brackets:

$20 \sin(20x) \sin(11x) = -10 \cos(31x) + 10 \cos(9x)$

Thus, we have successfully rewritten the product $20 \sin(20x) \sin(11x)$ as a difference of cosine functions. The rewritten expression is:

$20 \sin(20x) \sin(11x) = 10 \cos(9x) - 10 \cos(31x)$

This transformation allows us to express the product of two sine functions as a combination of cosine functions, which can be more convenient for further analysis or simplification. For instance, this form might be easier to integrate or differentiate, depending on the context.

Step-by-Step Solution

To summarize, here's a step-by-step solution for rewriting the product $20 \sin(20x) \sin(11x)$ as a sum or difference:

1. Identify the appropriate product-to-sum identity:
   • Since we have a product of two sine functions, we use the identity:
     • $\sin(A) \sin(B) = -\frac{1}{2} [\cos(A + B) - \cos(A - B)]$
2. Assign values to A and B:
   • In our expression, $20 \sin(20x) \sin(11x)$, we have $A = 20x$ and $B = 11x$.
3. Substitute A and B into the identity:
   • $\sin(20x) \sin(11x) = -\frac{1}{2} [\cos(20x + 11x) - \cos(20x - 11x)]$
4. Simplify the expression inside the brackets:
   • $\sin(20x) \sin(11x) = -\frac{1}{2} [\cos(31x) - \cos(9x)]$
5. Multiply by the constant factor (20 in this case):
   • $20 \sin(20x) \sin(11x) = 20 \times -\frac{1}{2} [\cos(31x) - \cos(9x)]$
   • $20 \sin(20x) \sin(11x) = -10 [\cos(31x) - \cos(9x)]$
6. Distribute the constant to each term inside the brackets:
   • $20 \sin(20x) \sin(11x) = -10 \cos(31x) + 10 \cos(9x)$
7. Rewrite the expression:
   • $20 \sin(20x) \sin(11x) = 10 \cos(9x) - 10 \cos(31x)$

By following these steps, we can systematically rewrite any product of sine or cosine functions as a sum or difference, which is a crucial skill in trigonometric manipulations. This step-by-step approach ensures clarity and accuracy in the transformation process. Understanding each step and the underlying principles allows for confident application of these identities in various contexts.

Examples and Applications

To further illustrate the utility of product-to-sum identities, let's consider a few more examples and their applications.

Example 1: Rewriting a product of cosine functions

Rewrite the expression $4 \cos(5x) \cos(3x)$ as a sum.

• Identity: $\cos(A) \cos(B) = \frac{1}{2} [\cos(A + B) + \cos(A - B)]$
• Assign values: $A = 5x$, $B = 3x$
• Substitute: $\cos(5x) \cos(3x) = \frac{1}{2} [\cos(5x + 3x) + \cos(5x - 3x)]$
• Simplify: $\cos(5x) \cos(3x) = \frac{1}{2} [\cos(8x) + \cos(2x)]$
• Multiply by the constant: $4 \cos(5x) \cos(3x) = 4 \times \frac{1}{2} [\cos(8x) + \cos(2x)]$
• Distribute: $4 \cos(5x) \cos(3x) = 2 [\cos(8x) + \cos(2x)]$
• Final Result: $4 \cos(5x) \cos(3x) = 2 \cos(8x) + 2 \cos(2x)$

This example demonstrates how to rewrite a product of cosine functions as a sum of cosine functions, following a similar step-by-step approach as before.

Example 2: Rewriting a product of sine and cosine functions

Rewrite the expression $6 \sin(7x) \cos(2x)$ as a sum or difference.

• Identity: $\sin(A) \cos(B) = \frac{1}{2} [\sin(A + B) + \sin(A - B)]$
• Assign values: $A = 7x$, $B = 2x$
• Substitute: $\sin(7x) \cos(2x) = \frac{1}{2} [\sin(7x + 2x) + \sin(7x - 2x)]$
• Simplify: $\sin(7x) \cos(2x) = \frac{1}{2} [\sin(9x) + \sin(5x)]$
• Multiply by the constant: $6 \sin(7x) \cos(2x) = 6 \times \frac{1}{2} [\sin(9x) + \sin(5x)]$
• Distribute: $6 \sin(7x) \cos(2x) = 3 [\sin(9x) + \sin(5x)]$
• Final Result: $6 \sin(7x) \cos(2x) = 3 \sin(9x) + 3 \sin(5x)$

This example illustrates the application of the product-to-sum identity involving a sine and a cosine function, resulting in a sum of sine functions.

Applications

1. Integration: Product-to-sum identities are invaluable in simplifying integrals involving products of trigonometric functions. For example, the integral $\int \sin(mx) \sin(nx) dx$ can be easily solved once the product is converted into a sum or difference.
2. Signal Processing: In signal processing, these identities are used to analyze and manipulate signals that are expressed as products of sinusoidal functions. They help in breaking down complex signals into simpler components.
3. Physics: In physics, particularly in the study of waves, these identities are used to analyze the interference of waves. The superposition of two waves can often be represented as a product of trigonometric functions, which can be simplified using these identities.
4. Simplification of Trigonometric Expressions: Product-to-sum identities can simplify complex trigonometric expressions, making them easier to work with in various mathematical contexts.

By exploring these examples and applications, we gain a deeper understanding of how product-to-sum identities are not only theoretical tools but also practical aids in solving real-world problems. The ability to transform trigonometric expressions allows us to approach complex problems with greater ease and efficiency.

Conclusion

In conclusion, rewriting products of trigonometric functions as sums or differences is a valuable technique in trigonometry, with wide-ranging applications in mathematics, physics, engineering, and signal processing. The product-to-sum identities provide a systematic way to perform these transformations, and a clear understanding of these identities is crucial for anyone working with trigonometric expressions.

In this article, we focused on rewriting the expression $20 \sin(20x) \sin(11x)$ using the appropriate product-to-sum identity. We demonstrated the step-by-step process, highlighting the importance of identifying the correct identity, substituting the appropriate values, and simplifying the resulting expression. We also provided additional examples to illustrate the versatility of these identities and their applications in various contexts.

Mastering the product-to-sum identities enables us to simplify complex expressions, solve trigonometric equations, and perform integration more efficiently. It also enhances our ability to analyze and manipulate signals and waves, which is essential in many scientific and engineering disciplines. By understanding and applying these identities, we can unlock new insights and solve challenging problems with greater confidence and accuracy. The product-to-sum identities, therefore, form a cornerstone of trigonometric manipulation and are an indispensable tool in the arsenal of any mathematician, scientist, or engineer. Understanding these concepts thoroughly is not just an academic exercise, but a practical skill that significantly enhances problem-solving capabilities in a variety of real-world scenarios.