Write the sum as a product: cos(10.16) – cos(9.56) .
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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Question
![**Example Problem: Write the sum as a product**
Given Expression:
\[ \cos(10.1b) - \cos(9.5b) \]
Solution:
To convert the difference of cosines into a product, we can use the trigonometric identity:
\[ \cos A - \cos B = -2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right) \]
In this case, \( A = 10.1b \) and \( B = 9.5b \).
Thus, applying the identity:
\[ \cos(10.1b) - \cos(9.5b) = -2 \sin \left(\frac{10.1b + 9.5b}{2}\right) \sin \left(\frac{10.1b - 9.5b}{2}\right) \]
Simplify the expressions inside the sine functions:
\[ \cos(10.1b) - \cos(9.5b) = -2 \sin \left(\frac{19.6b}{2}\right) \sin \left(\frac{0.6b}{2}\right) \]
\[ \cos(10.1b) - \cos(9.5b) = -2 \sin (9.8b) \sin (0.3b) \]
Hence, the difference of cosines \(\cos(10.1b) - \cos(9.5b)\) as a product is:
\[ \cos(10.1b) - \cos(9.5b) = -2 \sin (9.8b) \sin (0.3b) \]
This step-by-step transformation of the sum into a product uses fundamental trigonometric identities.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0e60132c-6df6-4596-95bf-3dd1dce70541%2F82aabf1d-848f-452e-80ef-d8f6d9d857f6%2Fhvsuu5l_processed.png&w=3840&q=75)
Transcribed Image Text:**Example Problem: Write the sum as a product**
Given Expression:
\[ \cos(10.1b) - \cos(9.5b) \]
Solution:
To convert the difference of cosines into a product, we can use the trigonometric identity:
\[ \cos A - \cos B = -2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right) \]
In this case, \( A = 10.1b \) and \( B = 9.5b \).
Thus, applying the identity:
\[ \cos(10.1b) - \cos(9.5b) = -2 \sin \left(\frac{10.1b + 9.5b}{2}\right) \sin \left(\frac{10.1b - 9.5b}{2}\right) \]
Simplify the expressions inside the sine functions:
\[ \cos(10.1b) - \cos(9.5b) = -2 \sin \left(\frac{19.6b}{2}\right) \sin \left(\frac{0.6b}{2}\right) \]
\[ \cos(10.1b) - \cos(9.5b) = -2 \sin (9.8b) \sin (0.3b) \]
Hence, the difference of cosines \(\cos(10.1b) - \cos(9.5b)\) as a product is:
\[ \cos(10.1b) - \cos(9.5b) = -2 \sin (9.8b) \sin (0.3b) \]
This step-by-step transformation of the sum into a product uses fundamental trigonometric identities.
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