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Arada [10]
3 years ago
11

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

Mathematics
1 answer:
TiliK225 [7]3 years ago
8 0
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)
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