Sum to Product Formulas

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

sin(a) + sin(b) =     2sin ( (a + b)/2 ) cos ( (a – b)/2 )

sin(a) – sin(b) =     2cos ( (a + b)/2 ) sin ( (a – b)/2 )

cos(a) + cos(b) =    2cos ( (a + b)/2 ) cos ( (a – b)/2 )

cos(a) – cos(b) = – 2sin ( (a + b)/2 ) sin ( (a – b)/2 )

Example: Express cos(285) + cos(165) as a product and find the exact value.

Solution:

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

cos(a) + cos(b) = 2cos((a + b)/2)cos((a – b)/2)

cos(285) + cos(165) = 2cos(450/2)cos(120/2)

Step 2: Simplify and evaluate:

2cos(450/2)cos(120/2)

= 2cos(225)cos(60)

= 2(-√2/2)(1/2) = -√2/2

The exact value is -√2/2.

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