Question

Find a, b , and h so that f(x)= a sin (b(x-h))
f(x)=5cosx+2sinx

Answer 1

`a\sin(b(x-h))=a\sin(bx)\cos(bh)-a\cos(bx)\sin(bh)`

For this to be equivalent to
`5\cos x+2\sin x`, you require
`b=1` and


`\begin{cases}a\cos h=2\\-a\sin h=5\end{cases}`

Dividing the second equation by the first gives


`(-a\sin h)/(a\cos h)=-\tan h=\frac52\implies h=\arctan\left(-\frac52\right)=-\arctan\frac52`

Meanwhile, you also get


`a\cos\left(-\arctan\frac52\right)=2\implies a\cos\left(\arctan\frac52\right)=(2a)/(√(29))=2\implies a=√(29)`

So,


`f(x)=5\cos x+2\sin x=√(29)\sin\left(x+\arctan\frac52\right)`