Use the sum-to-product formula to fill in the blanks in the identity below: sin(x) + sin(x) = 2 sin(7x) cos(3x) Note: You can earn partial credit on this problem.
Use the sum-to-product formula to fill in the blanks in the identity below: sin(x) + sin(x) = 2 sin(7x) cos(3x) Note: You can earn partial credit on this problem.
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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![**Using the Sum-to-Product Formula in Trigonometry**
**Problem Statement:**
Use the sum-to-product formula to fill in the blanks in the identity below:
\[ \sin(\square x) + \sin(\square x) = 2\sin(7x)\cos(3x) \]
**Note:** You can earn partial credit on this problem.
**Explanation for Students:**
To solve this problem, you need to use the sum-to-product identities in trigonometry. These identities convert sums of sines or cosines into products, which can be easier to work with in many cases.
The sum-to-product formula for sine is:
\[ \sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \]
Given the right-hand side, \( 2\sin(7x)\cos(3x) \), we identify parts of our formula:
1. **Identify \( A \) and \( B \):**
- \( A = 10x \)
- \( B = 4x \)
2. **Substitute \( A \) and \( B \) into the sum-to-product formula:**
- \( \sin A + \sin B \)
- \( 2 \sin \left( \frac{10x + 4x}{2} \right) \cos \left( \frac{10x - 4x}{2} \right) \)
3. **Simplify the expressions:**
- \( A + B = 10x + 4x = 14x \)
- \( \frac{14x}{2} = 7x \)
- \( A - B = 10x - 4x = 6x \)
- \( \frac{6x}{2} = 3x \)
So this confirms the conversion:
\[ \sin(10x) + \sin(4x) = 2\sin(7x)\cos(3x) \]
Thus, the blanks can be filled in as follows:
\[ \sin(10x) + \sin(4x) = 2\sin(7x)\cos(3x) \]
Understanding and implementing these formulas are crucial not only for simplifying expressions but also for solving](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e08cb41-5b2e-4118-8784-a24243109431%2F7077b076-7697-4366-996e-772c8960df09%2Fvenoodi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Using the Sum-to-Product Formula in Trigonometry**
**Problem Statement:**
Use the sum-to-product formula to fill in the blanks in the identity below:
\[ \sin(\square x) + \sin(\square x) = 2\sin(7x)\cos(3x) \]
**Note:** You can earn partial credit on this problem.
**Explanation for Students:**
To solve this problem, you need to use the sum-to-product identities in trigonometry. These identities convert sums of sines or cosines into products, which can be easier to work with in many cases.
The sum-to-product formula for sine is:
\[ \sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \]
Given the right-hand side, \( 2\sin(7x)\cos(3x) \), we identify parts of our formula:
1. **Identify \( A \) and \( B \):**
- \( A = 10x \)
- \( B = 4x \)
2. **Substitute \( A \) and \( B \) into the sum-to-product formula:**
- \( \sin A + \sin B \)
- \( 2 \sin \left( \frac{10x + 4x}{2} \right) \cos \left( \frac{10x - 4x}{2} \right) \)
3. **Simplify the expressions:**
- \( A + B = 10x + 4x = 14x \)
- \( \frac{14x}{2} = 7x \)
- \( A - B = 10x - 4x = 6x \)
- \( \frac{6x}{2} = 3x \)
So this confirms the conversion:
\[ \sin(10x) + \sin(4x) = 2\sin(7x)\cos(3x) \]
Thus, the blanks can be filled in as follows:
\[ \sin(10x) + \sin(4x) = 2\sin(7x)\cos(3x) \]
Understanding and implementing these formulas are crucial not only for simplifying expressions but also for solving
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