Express the given product as a sum containing only sines or cosines. cos (20) cos (80) *** cos (20) cos (80) = (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
Express the given product as a sum containing only sines or cosines. cos (20) cos (80) *** cos (20) cos (80) = (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
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
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### Trigonometric Product to Sum Conversion
#### Problem Statement:
**Express the given product as a sum containing only sines or cosines:**
\[ \cos(2\theta) \cos(8\theta) \]
---
#### Solution Box:
\[ \cos(2\theta) \cos(8\theta) = \text{[Your Answer Here]} \]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
---
### Explanation:
In this exercise, we will convert the product of two cosine functions into a sum involving only sine or cosine functions. This type of problem often requires the use of trigonometric identities such as product-to-sum identities.
The product-to-sum identities for cosine are given by:
\[ \cos(A) \cos(B) = \frac{1}{2} [ \cos(A - B) + \cos(A + B) ] \]
Using this identity, let \( A = 2\theta \) and \( B = 8\theta \):
\[ \cos(2\theta) \cos(8\theta) = \frac{1}{2} [ \cos(2\theta - 8\theta) + \cos(2\theta + 8\theta) ] \]
Simplify the angles inside the cosine functions:
\[ \cos(2\theta) \cos(8\theta) = \frac{1}{2} [ \cos(-6\theta) + \cos(10\theta) ] \]
Using the property that \( \cos(-x) = \cos(x) \):
\[ \cos(2\theta) \cos(8\theta) = \frac{1}{2} [ \cos(6\theta) + \cos(10\theta) ] \]
Therefore, the final expression for \( \cos(2\theta) \cos(8\theta) \) as a sum of cosines is:
\[ \cos(2\theta) \cos(8\theta) = \frac{1}{2} \cos(6\theta) + \frac{1}{2} \cos(10\theta) \]
You can write your final answer in the provided solution box.
---
Note: Ensure that you enter any simplified terms with any radicals, if present, and use integers or fractions for any numerical values in the final expression](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff11f2156-79e1-4988-b339-1b50961c547d%2F8fa90007-101b-4c3c-9339-c8cdd122eefe%2Fjj1ejdi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Trigonometric Product to Sum Conversion
#### Problem Statement:
**Express the given product as a sum containing only sines or cosines:**
\[ \cos(2\theta) \cos(8\theta) \]
---
#### Solution Box:
\[ \cos(2\theta) \cos(8\theta) = \text{[Your Answer Here]} \]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
---
### Explanation:
In this exercise, we will convert the product of two cosine functions into a sum involving only sine or cosine functions. This type of problem often requires the use of trigonometric identities such as product-to-sum identities.
The product-to-sum identities for cosine are given by:
\[ \cos(A) \cos(B) = \frac{1}{2} [ \cos(A - B) + \cos(A + B) ] \]
Using this identity, let \( A = 2\theta \) and \( B = 8\theta \):
\[ \cos(2\theta) \cos(8\theta) = \frac{1}{2} [ \cos(2\theta - 8\theta) + \cos(2\theta + 8\theta) ] \]
Simplify the angles inside the cosine functions:
\[ \cos(2\theta) \cos(8\theta) = \frac{1}{2} [ \cos(-6\theta) + \cos(10\theta) ] \]
Using the property that \( \cos(-x) = \cos(x) \):
\[ \cos(2\theta) \cos(8\theta) = \frac{1}{2} [ \cos(6\theta) + \cos(10\theta) ] \]
Therefore, the final expression for \( \cos(2\theta) \cos(8\theta) \) as a sum of cosines is:
\[ \cos(2\theta) \cos(8\theta) = \frac{1}{2} \cos(6\theta) + \frac{1}{2} \cos(10\theta) \]
You can write your final answer in the provided solution box.
---
Note: Ensure that you enter any simplified terms with any radicals, if present, and use integers or fractions for any numerical values in the final expression
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