Question

How do you change `2 \sin 7x \cos 4x` into a sum?

Answer 1

sin 11x + sin 3x

Explanation:

`f(x) = 2cos 4x. sin 7x`
Note that `sin 7x = cos (pi/2 - 7x)` (complementary arcs)
`f(x) = 2 cos 4x.cos (pi/2 - 7x)`
Reminder of trig identity:
`2cos a.cos b = cos (a - b) + cos (a + b)`
In this case:
`a - b = 4x - (pi/2 - 7x) = 11x - pi/2`
`a + b = 4x + pi/2 - 7x = pi/2 - 3x`
`f(x) = cos (11x - pi2) + cos (pi/2 - 3x) = cos (pi/2 - 11x) + sin 3x`
`f(x) = sin 11x + sin 3x` (complementary arcs)