7.5 Product-to-Sum and Sum-to-Product Formulas

7.5 Product-to-Sum and Sum-to-Product Formulas

This section is one of the more basic trigonometric concepts, changing products to sums or sums to products.

The proof of the product-to-sum formulas is:

                     sin (u + v) = sin u cos v + cos u sin v
                     sin (u + v) = sin u cos v - cos u sin v
sin (u + v) + sin (u - v) = 2 sin u cos v

This allows us to use these formulas:

(1) sin u cos v = 1/2[sin (u + v) (u - v)]
(2) cos u sin v = 1/2[sin (u + v) (u - v)]
(3) cos u cos v = 1/2[cos (u + v) (u - v)]
(4) sin u sin v = 1/2[cos (u - v) - cos (u + v)]


The proof of the sum-to-product formulas is:

                                           1) u + v = a       and        u - v = b
2) (u + v) + (u - v) = a + b                                                      2) (u + v) - (u - v) = a - b              
3) u = a + b                                                                              3) v = a - b
               2                                                                                               2

         Substitute for u + v and u - v on the right-hand sides of the product-to-sum formulas and for u and v on the left-hand sides. Multiply by 2 and we obtain the following sum-to-product formulas.

(1) sin a + sin b = 2 sin a + b cos a - b

                                           2             2

(2) sin a - sin b = 2 cos a + b sin a - b

                                          2            2

(3) cos a + cos b = 2 cos a + b cos a - b

                                             2             2

(4) cos a - cos b = -2 sin a + b sin a - b

                                            2             2

There are many tremendous youtube videos on this topic. I checked.

Peace.Love.Thad

-Joey