Question

How do you use the transformation formulas to go from product to sum and sum to product?

Answer 1

Main approach to solve a trig equation : Use Trig Transformation Identities to transform it to a product of a few basic trig equations. Solving a trig equation finally results in solving a few basic trig equations.
Transformation Trig Identities that convert Sums to Products .
1. cos a + cos b = 2cos (a +b)/2cos (a - b)/2
2. cos a - cos b = -2sin (a + b)/2sin (a - b)/2
3. sin a + sin b = 2sin (a + b)/2cos (a - b)/2
4. sin a - sin b = 2cos (a + b)/2sin (a - b)/2
5. tan a + tan b = sin (a + b)/cos acos b.
6. tan a - tan b = sin (a - b)/cos acos b
Example 1 . Transform f(x) = sin a + cos a to a product.
Solution. Use Identity (3) to transform f(x) = sin a + sin (Pi/2 - a) = 2sin (Pi/4)sin (a + Pi/4)
Example 2 . Transform f(x) = sin x + sin 3x + sin 2x to a product. Use Identity (3) to transform the sum (sin x + sin 3x), then put in common factor.
f(x) = (2sin 2acos a) + 2sin acos a = 2cos a(2sin 3a/2*cos a/2)