Question

Find A and B if sin2xcos3x = A sin 5x + B sin x

Answer 1

Recall that


`\sin(x+y)=\sin x\cos y+\cos x\sin y`

`\sin(x-y)=\sin x\cos y-\cos x\sin y`

Adding these together, you have


`\sin(x+y)+\sin(x-y)=2\sin x\cos y`

`\frac12\sin(x+y)+\frac12\sin(x-y)=\sin x\cos y`

Replace
`x` with
`2x` and
`y` with
`3x`. You end up with


`\frac12\sin(2x+3x)+\frac12\sin(2x-3x)=\sin2x\cos3x`

`\sin2x\cos3x=\frac12\sin5x+\frac12\sin(-x)=\frac12\sin5x-\frac12\sin x`

and so
`A=\frac12` and
`B=-\frac12`.