Sum-to-Product Formulas

Trigonometry · Axiom Academy

LESSON Sum-to-Product Formulas Transform sums and differences of trig functions into products—a powerful technique for factoring and simplification 1. Why Convert Sums to Products? Consider the expression sin(5x) + sin(3x). As a sum, this is difficult to factor or simplify. But what if we could write it as a product of simpler functions? 2. The Four Sum-to-Product Formulas These formulas convert sums and differences into products. Notice the pattern: the arguments become averages and half-differences. 3. Worked Example: Factoring sin(5x) + sin(3x) Let's apply the sum-to-product formula step by step to convert this sum into a factored product form. Step 1 Identify A = 5x and B = 3x Step 2 Apply the formula: sin(A) + sin(B) = 2sin((A+B)/2)cos((A-B)/2) Step 3 Calculate (A+B)/2 = (5x+3x)/2 = 4x Step 4 Calculate (A-B)/2 = (5x-3x)/2 = x Result sin(5x) + sin(3x) = 2sin(4x)cos(x) 4. When to Use Sum-to-Product Formulas These formulas are particularly useful in specific situations. Recognizing when to apply them is key to efficient problem solving. Solve equations: sin(3x) + sin(x) = 0 becomes 2sin(2x)cos(x) = 0, giving two simpler equations. Simplify fractions: Factor numerator and denominator to cancel common terms. Prove identities: Convert one side to match the other by factoring strategically. Integration: Products often integrate more easily than sums, especially with substitution.

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