Product to Sum Formulas

Definition: Also known as product to sum identities, these formulas allow us to rewrite sine and cosine products as sums or differences.

cos(a)cos(b) = ½(cos(a + b) + cos(a – b))

sin(a)sin(b) = ½(cos(a – b) – cos(a + b))

sin(a)cos(b) = ½(sin(a + b) + sin(a – b))

cos(a)sin(b) = ½(sin(a + b) – sin(a – b))

Example: Express cos(75)sin(195) as a sum or difference and find the exact value.

Solution:

Step 1: Determine the best formula to use and plug in the given values:

cos(a)sin(b) = ½(sin(a + b) – sin(a – b))

cos(75)sin(195) = ½(sin(270) – sin(-120))

Step 2: Evaluate and simplify to find the exact value:

½(sin(270) – sin(-120))

= (1/2)(√2/2) + (√3/2)(√2/2)

= 1/2(-1 – (-√3/2)

The exact value is -1/2 + √3/4.

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