Solve sin ( 4 x ) cos ( 6 x ) − cos ( 4 x ) sin ( 6 x ) = − 0.2 for the smallest positive solution.

Answer 1

If x is small and positive you can Taylor expand the functions sin(x) and cos(x) about x=0.

In this case you have

sin(4x) ∼ 4x

sin(6x) ∼ 6x

cos(4x) ∼ 1

cos(6x) ∼1

Put the above terms in your original equation gives

4x-6x=-0.2 which means that x=0.1

Best,

Davide