Converting 2sin(3x)cos(x) to Sum

Trigonometry · Axiom Academy

EXAMPLE Product-to-Sum Formula Convert 2sin(3x)cos(x) to sum form using the product-to-sum identity Excellent work! You've successfully converted a product to sum form. Here's what we learned: Pattern Recognition: The expression 2sin(A)cos(B) matches the product-to-sum formula perfectly. Identifying this pattern is the first crucial step. Identify Variables: In 2sin(3x)cos(x), we identified A = 3x and B = x. These values are then used throughout the formula application. Apply the Formula: The product-to-sum formula 2sin(A)cos(B) = sin(A+B) + sin(A-B) transforms products into sums that are often easier to work with. Simplify Carefully: After substitution, simplify A+B and A-B step by step: (3x + x) = 4x and (3x - x) = 2x. Final Form: The result sin(4x) + sin(2x) is the sum form, which can be more convenient for integration, solving equations, or further manipulation. This product-to-sum technique is invaluable in trigonometry! You'll use similar formulas for cos(A)cos(B), sin(A)sin(B), and other combinations. Master this process and you'll breeze through trigonometric simplifications!

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