Question

c2*sin(wt)+c1*cos(wt)= A*Sin(wt+phi), where c2=Acos(Phi) and c1=Asin(Phi). They ask me to find the amplitude of the function 2 sin(4pi*t)+5 cos(4pi*t), in terms of A sin(wt+phi). How do i do this?

Answer 1

The amplitude of $A\sin(\omega t+\phi)$ is the absolute value of $A$. So first you need to condense the given function into one sine expression.

Recall that

$A\sin(\omega t+\phi)=A\sin(\omega t)\cos\phi+A\cos(\omega t)\sin\phi$

so you need to choose $\phi$ and $\omega$ accordingly.

If we line up the terms of the given function with the expanded one above, we should have

$2\sin(4\pi t)+5\cos(4\pi t)\implies\begin{cases}A\cos\phi=2\\A\sin\phi=5\\\omega=4\pi\end{cases}$

Now, using the Pythagorean identity,

$(A\sin\phi)^2+(A\cos\phi)^2=2^2+5^2\implies A^2=29\implies A=\pm\sqrt{29}$

so the amplitude is √29.

Just for completeness, we also get

$\tan\phi=\dfrac{\sin\phi}{\cos\phi}=\dfrac52\implies\phi=\tan^{-1}\left(\dfrac52\right)+n\pi$

where $n$ is any integer.