https://r2.erweima.ai/mathbot/image/1723681566065rsfjl7wd.jpg

Write the sum of this equation as a product 

To express the difference of cosines as a product, we can use the following trigonometric identity:

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

Given the expression \(\cos(15.4a) - \cos(9.2a)\), we can apply the identity:

1. **Identify** \(A = 15.4a\) and \(B = 9.2a\).
2. **Calculate** the average and difference of the angles:

   \[
   \frac{A+B}{2} = \frac{15.4a + 9.2a}{2} = \frac{24.6a}{2} = 12.3a
   \]
   
   \[
   \frac{A-B}{2} = \frac{15.4a - 9.2a}{2} = \frac{6.2a}{2} = 3.1a
   \]

3. **Substitute** these into the identity:

   \[
   \cos(15.4a) - \cos(9.2a) = -2 \sin(12.3a) \sin(3.1a)
   \]

So, the expression as a product is:

\[
\boxed{-2 \sin(12.3a) \sin(3.1a)}
\]

Would you like any further details or explanations? Here are some related questions you might find interesting:

1. How do you convert a sum of sines into a product?
2. What other trigonometric identities involve sums or differences?
3. Can you derive the product-to-sum identities from the sum-to-product identities?
4. What is the general form for the difference of two cosines?
5. How can the identity used above be applied in solving trigonometric equations?
6. How do you express \(\cos(A) + \cos(B)\) as a product?
7. What are the conditions under which the sum-to-product identities are most useful?
8. How can this identity be applied to real-world problems in physics or engineering?

**Tip:** When working with trigonometric identities, it's helpful to remember the most commonly used identities, such as sum-to-product and product-to-sum, as they often simplify complex expressions.

Explore how to transform the difference of cosines into a product using trigonometric identities. Learn step-by-step with detailed calculations and examples. Suitable for high school students and anyone interested in trigonometry and mathematical identities.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation