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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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.3: Trigonometric Functions Of Real Numbers
Problem 21E
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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
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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