Question

Write the product as a sum: 14cos(39x)sin(19x)

Answer 1

Recall that

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

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

Adding these together gives

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

To get 14 cos(39x) sin(19x) on the right side, multiply both sides by 7 and replace a = 19x and b = 39x :

7 (sin(19x + 39x) + sin(19x - 39x)) = 14 cos(39x) sin(19x)

7 (sin(58x) + sin(-20x)) = 14 cos(39x) sin(19x)

7 (sin(58x) - sin(20x)) = 14 cos(39x) sin(19x)