Answered: Write the sum as a product: cos(24.96)…

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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$## Converting Trigonometric Expressions ### Problem Statement Write the sum as a product: \[ \cos(24.9t) - \cos(12.7t) = \] ### Explanation The image shows a mathematical problem where you need to transform a difference of cosines into a product form. This can typically be achieved using trigonometric identities. Using the product-to-sum identities for cosine, we know: \[ \cos(A) - \cos(B) = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \] Here, $ A $ is $ 24.9t $ and $ B $ is $ 12.7t $. #### Steps: 1. **Calculate $ A + B $ and divide by 2** \[ \frac{24.9t + 12.7t}{2} = \frac{37.6t}{2} = 18.8t \] 2. **Calculate $ A - B $ and divide by 2** \[ \frac{24.9t - 12.7t}{2} = \frac{12.2t}{2} = 6.1t \] 3. **Substitute into the identity** \[ \cos(24.9t) - \cos(12.7t) = -2 \sin(18.8t) \sin(6.1t) \] ### Conclusion Therefore, the difference of cosines $ \cos(24.9t) - \cos(12.7t) $ can be written as the product: \[ \cos(24.9t) - \cos(12.7t) = -2 \sin(18.8t) \sin(6.1t) \] Fill in the product in the provided blank box for the transformation. This exercise helps students to understand and utilize trigonometric identities for converting sums or differences of trigonometric functions into products.$

Transcribed Image Text:## Converting Trigonometric Expressions ### Problem Statement Write the sum as a product: \[ \cos(24.9t) - \cos(12.7t) = \] ### Explanation The image shows a mathematical problem where you need to transform a difference of cosines into a product form. This can typically be achieved using trigonometric identities. Using the product-to-sum identities for cosine, we know: \[ \cos(A) - \cos(B) = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \] Here, $ A $ is $ 24.9t $ and $ B $ is $ 12.7t $. #### Steps: 1. **Calculate $ A + B $ and divide by 2** \[ \frac{24.9t + 12.7t}{2} = \frac{37.6t}{2} = 18.8t \] 2. **Calculate $ A - B $ and divide by 2** \[ \frac{24.9t - 12.7t}{2} = \frac{12.2t}{2} = 6.1t \] 3. **Substitute into the identity** \[ \cos(24.9t) - \cos(12.7t) = -2 \sin(18.8t) \sin(6.1t) \] ### Conclusion Therefore, the difference of cosines $ \cos(24.9t) - \cos(12.7t) $ can be written as the product: \[ \cos(24.9t) - \cos(12.7t) = -2 \sin(18.8t) \sin(6.1t) \] Fill in the product in the provided blank box for the transformation. This exercise helps students to understand and utilize trigonometric identities for converting sums or differences of trigonometric functions into products.

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