Question

Write y(t)=2sin 4 pi t + 5 cos 4 pi t) in the form y(t) A sin (wt + Ø) and identify the amplitude, angular frequency, and the phase shift of the spring motion.

Record your answers in the response box.

Answer 1

Expanding the desired form, we have

`A \sin(\omega t + \phi) = A \bigg(\sin(\omega t) \cos(\phi) + \cos(\omega t) \sin(\phi)\bigg)`

and matching it up with the given expression, we see that

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

A natural choice for one of the symbols is
`\omega = 4\pi`. Then

`\begin{cases} A \cos(\phi) = 2 \\ A \sin(\phi) = 5 \end{cases}`

Use the Pythagorean identity to eliminate
`\phi`.

`(A\cos(\phi))^2 + (A\sin(\phi))^2 = A^2 \cos^2(\phi) + A^2 \sin^2(\phi) = A^2 (\cos^2(\phi) + \sin^2(\phi)) = A^2`

so that

`A^2 = 2^2 + 5^2 = 29 \implies A = \pm√(29)`

Use the definition of tangent to eliminate
`A`.

`\tan(\phi) = (\sin(\phi))/(\cos(\phi)) = (A\sin(\phi))/(\cos(\phi))`

so that

`\tan(\phi) = \frac52 \implies \phi = \tan^(-1)\left(\frac52\right)`

We end up with

`y(t) = 2 \sin(4\pi t) + 5 \cos(4\pi t) = \boxed{\pm√(29) \sin\left(4\pi t + \tan^(-1)\left(\frac52\right)\right)}`

where

• amplitude:

`|A| = \boxed{√(29)}`

• angular frequency:

`\boxed{4\pi}`

• phase shift:

`4\pi t + \tan^(-1)\left(\frac 52\right) = 4\pi \left(t + \boxed{\frac1{4\pi} \tan^(-1)\left(\frac52\right)}\,\right)`