Write the product as a sum or difference:

Question

Write the product as a sum or difference:

 

Answer 1

You can either derive this by noticing that the sin addition formula includes sin and cos, but two terms, so we will use the sum of two sins to eliminate one of the terms, or just look it up.

sin(10a + 15a) = sin(10a) cos(15a) + cos(10a) sin(15a). As we wish to remove the second term, and cos is even but sin is odd, we add

sin(10a - 15a) = sin(10a) cos(-15a) + cos(10a) sin(-15a) = sin(10a) cos(15a) - cos(10a) sin(15a)

Thus, sin(10a + 15a) + sin(10a - 15a) = 2 sin(10a) cos(15a)

sin(25a) + sin(-5a) = 2 sin(10a) cos(15a), or

sin(25a) - sin(5a) = 2 sin(10a) cos(15a)

The original problem has 18 cos (15a) sin (10 a), so the solution is

9(2 sin(10a) cos(15a)) =

9 sin(25a) - 9 sin(5a)