# Laplace Transformation question

• Nov 28th 2012, 12:41 AM
joshua111
Laplace Transformation question
Hi, i do not know where to post this so i just figured i leave this where there are most people viewing. I need help doing question 4 only. Thanks!!
Attachment 25965
Attachment 25966
• Nov 28th 2012, 02:43 AM
BobP
Re: Laplace Transformation question
The steady state part can be expressed in the form

$\displaystyle v_{steady}=A\sin (\omega t + \alpha) = A\sin \omega t \cos \alpha + A \cos \omega t \sin \alpha,$

where $\displaystyle A$ is the amplitude.

Equate coefficients of $\displaystyle \sin \omega t$ and $\displaystyle \cos \omega t$ with the corresponding expressions in your formula for $\displaystyle v_{steady},$ and eliminate the $\displaystyle \cos \alpha \text{ and } \ sin \alpha$ by squaring and adding.
• Nov 28th 2012, 02:50 AM
joshua111
Re: Laplace Transformation question
Quote:

Originally Posted by BobP
The steady state part can be expressed in the form

$\displaystyle v_{steady}=A\sin (\omega t + \alpha) = A\sin \omega t \cos \alpha + A \cos \omega t \sin \alpha,$

where $\displaystyle A$ is the amplitude.

Equate coefficients of $\displaystyle \sin \omega t$ and $\displaystyle \cos \omega t$ with the corresponding expressions in your formula for $\displaystyle v_{steady},$ and eliminate the $\displaystyle \cos \alpha \text{ and } \ sin \alpha$ by squaring and adding.

Hi,
I'm really bad at this but if you have the time, do you mind showing how to get to the final answer?
Thanks!
• Nov 28th 2012, 05:30 AM
BobP
Re: Laplace Transformation question
If $\displaystyle A\sin \theta = p \text{ and } A\cos \theta = q,$
$\displaystyle A^{2}(\sin ^{2}\theta + \cos ^{2}\theta) = p^{2} + q^{2},$
so
$\displaystyle A = \sqrt {p^{2} + q^{2}.$
And sorry, but I know absolutely nothing about the filters mentioned in the question.
• Nov 28th 2012, 08:04 AM
joshua111
Re: Laplace Transformation question
Quote:

Originally Posted by BobP
If $\displaystyle A\sin \theta = p \text{ and } A\cos \theta = q,$
$\displaystyle A^{2}(\sin ^{2}\theta + \cos ^{2}\theta) = p^{2} + q^{2},$
$\displaystyle A = \sqrt {p^{2} + q^{2}.$
I didn't derive anything in post 2, I simply stated what should (at your stage) be a well-known trig identity, and what allows the the expression for $\displaystyle v_{steady}$ given in the question to be written in an alternative form.
Another trig identity, with which you should be familiar, is used later. $\displaystyle \cos^{2}\theta+\sin^{2}\theta=1.$