Write the product as a sum: 20 sin(23u)sin(9v)
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°.
Related questions
Question
![### Converting a Product to a Sum
The following task involves rewriting a trigonometric product as a sum using trigonometric identities.
**Problem Statement:**
Write the product as a sum:
\[ 20 \sin(32x) \sin(9x) = \]
**Explanation:**
To express the product of sine functions as a sum, we can use the product-to-sum identities in trigonometry:
\[ \sin(A)\sin(B) = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \]
For our problem:
- \( A = 32x \)
- \( B = 9x \)
Applying the identity:
\[ \sin(32x)\sin(9x) = \frac{1}{2} [\cos(32x - 9x) - \cos(32x + 9x)] \]
\[ = \frac{1}{2} [\cos(23x) - \cos(41x)] \]
Now, multiplying by 20:
\[ 20 \sin(32x) \sin(9x) = 20 \cdot \frac{1}{2} [\cos(23x) - \cos(41x)] \]
\[ = 10 [\cos(23x) - \cos(41x)] \]
Therefore:
\[ 20 \sin(32x) \sin(9x) = 10 \cos(23x) - 10 \cos(41x) \]
**Final Answer:**
\[ 20 \sin(32x) \sin(9x) = 10 \cos(23x) - 10 \cos(41x) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8cdbc022-1ecb-430f-8c37-0abecd7b3ab0%2F18e64715-289e-4db4-bb0b-e28c4534030b%2F0tay6o_processed.png&w=3840&q=75)
Transcribed Image Text:### Converting a Product to a Sum
The following task involves rewriting a trigonometric product as a sum using trigonometric identities.
**Problem Statement:**
Write the product as a sum:
\[ 20 \sin(32x) \sin(9x) = \]
**Explanation:**
To express the product of sine functions as a sum, we can use the product-to-sum identities in trigonometry:
\[ \sin(A)\sin(B) = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \]
For our problem:
- \( A = 32x \)
- \( B = 9x \)
Applying the identity:
\[ \sin(32x)\sin(9x) = \frac{1}{2} [\cos(32x - 9x) - \cos(32x + 9x)] \]
\[ = \frac{1}{2} [\cos(23x) - \cos(41x)] \]
Now, multiplying by 20:
\[ 20 \sin(32x) \sin(9x) = 20 \cdot \frac{1}{2} [\cos(23x) - \cos(41x)] \]
\[ = 10 [\cos(23x) - \cos(41x)] \]
Therefore:
\[ 20 \sin(32x) \sin(9x) = 10 \cos(23x) - 10 \cos(41x) \]
**Final Answer:**
\[ 20 \sin(32x) \sin(9x) = 10 \cos(23x) - 10 \cos(41x) \]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, trigonometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning