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

$\dfrac{-a\sin h}{a\cos h}=-\tan h=\dfrac52\implies h=\arctan\left(-\dfrac52\right)=-\arctan\dfrac52$

Meanwhile, you also get

$a\cos\left(-\arctan\dfrac52\right)=2\implies a\cos\left(\arctan\dfrac52\right)=\dfrac{2a}{\sqrt{29}}=2\implies a=\sqrt{29}$

So,

$f(x)=5\cos x+2\sin x=\sqrt{29}\sin\left(x+\arctan\dfrac52\right)$