Write the sum as a product: cos(12.7z) + cos(3.5z)
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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![### Write the Sum as a Product
Given the trigonometric expression:
\[ \cos(12.7x) + \cos(3.5x) \]
**Task:** Rewrite the sum as a product.
To solve this problem, we use the sum-to-product identities in trigonometry. The sum-to-product identities allow us to express sums of sines and cosines as products of sines and cosines. For cosines, this identity is:
\[ \cos(A) + \cos(B) = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
Applying this formula to our given expression:
1. Let \( A = 12.7x \)
2. Let \( B = 3.5x \)
Substituting \( A \) and \( B \) into the formula:
\[ \cos(12.7x) + \cos(3.5x) = 2 \cos\left(\frac{12.7x + 3.5x}{2}\right) \cos\left(\frac{12.7x - 3.5x}{2}\right) \]
Simplify the expressions inside the cosines:
\[ = 2 \cos\left(\frac{16.2x}{2}\right) \cos\left(\frac{9.2x}{2}\right) \]
\[ = 2 \cos(8.1x) \cos(4.6x) \]
So the sum:
\[ \cos(12.7x) + \cos(3.5x) \]
Can be written as the product:
\[ 2 \cos(8.1x) \cos(4.6x) \]
This conversion is useful in various mathematical and engineering applications where product forms can simplify the problem at hand or facilitate further analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0e60132c-6df6-4596-95bf-3dd1dce70541%2F415fdfd0-ef32-461e-ad0b-24b5f0767cfa%2Fqn5omkg_processed.png&w=3840&q=75)
Transcribed Image Text:### Write the Sum as a Product
Given the trigonometric expression:
\[ \cos(12.7x) + \cos(3.5x) \]
**Task:** Rewrite the sum as a product.
To solve this problem, we use the sum-to-product identities in trigonometry. The sum-to-product identities allow us to express sums of sines and cosines as products of sines and cosines. For cosines, this identity is:
\[ \cos(A) + \cos(B) = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
Applying this formula to our given expression:
1. Let \( A = 12.7x \)
2. Let \( B = 3.5x \)
Substituting \( A \) and \( B \) into the formula:
\[ \cos(12.7x) + \cos(3.5x) = 2 \cos\left(\frac{12.7x + 3.5x}{2}\right) \cos\left(\frac{12.7x - 3.5x}{2}\right) \]
Simplify the expressions inside the cosines:
\[ = 2 \cos\left(\frac{16.2x}{2}\right) \cos\left(\frac{9.2x}{2}\right) \]
\[ = 2 \cos(8.1x) \cos(4.6x) \]
So the sum:
\[ \cos(12.7x) + \cos(3.5x) \]
Can be written as the product:
\[ 2 \cos(8.1x) \cos(4.6x) \]
This conversion is useful in various mathematical and engineering applications where product forms can simplify the problem at hand or facilitate further analysis.
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