I am currently prepping for uni having been a few years out of the studying loop (programming as it happens). Anyway, I've been re-reading my A-level notes/exercises and looking through OpenCourseWare stuff and I noticed this assertation in my notes:
Given $a \cos\theta \pm b \sin\theta = c$ then:
$a \sin\theta \pm b\cos\theta = r\sin(\theta\pm\alpha)+c$
$a \cos\theta \pm b\sin\theta = r\cos(\theta\pm\alpha)+c$
Given $r \in \mathbb{N}$, $0 \leq \theta < 2\pi$.
Now, I've heard of fourier series which have a very similar form to these equestions.
So, my question is, is there a relation between the two?
Please bear in mind I know I'm stepping off a cliff and into "unknown unknowns" territory if they are – I have absolutely no idea what harmonic analysis is and I don't (yet) understand fourier series fully, although I grasp roughly how they work. I'm just interested to know if they are related and of course any further information / direction / reading I should follow up. I ask because this was one of those "odd topics" at A-level that we derived in class from a geometric argument then I've never seen again.
Thanks.
Best Answer
I can't really make sense of those equations, but it looks something like the statement that for any $ a, b \in \mathbb{R} $, then for $ r = \sqrt{a^2 + b^2 }$ and $ \phi = \arctan(b/a) $ we have
$ a \sin(\theta) + b \cos(\theta) = r \sin(\theta + \phi).$
If you visualize x and y axes with $ cos(\theta) $ along the x-axis and $ sin(\theta) $ along the y-axis, then this says we can identify any sinusoidal, i.e. any function like $ r \sin(\theta + \phi) $, by its cartesian coordinates $(a,b)$ in this plane, or its polar coordinate $(r, \phi )$. $r$ is called the magnitude of the sinusodial, and $ \phi $ the phase.
This is directly related to Fourier Analysis. The main idea in Fourier Analysis is to decompose a function into its sinusodial components. For instance, if $f$ is a real-valued function on the interval $ [0, 2\pi] $ that is suitably regular, it turns out you can write $f$ uniquely as an infinite sum of sine and cosines, called the Fourier Series:
$ f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx)). $
The numbers $ a_n $ and $ b_n $ are called the Fourier coefficients of your function.
I think this representation of Fourier series is sometimes difficult to grasp. But using the relationship above, we see we could have also wrote:
$ f(x) = a_0 + \sum_{n=1}^{\infty} r_n \sin(nx + \phi_n) $
In this form I think you can see more clearly what's going on: For each $n$, there is only one sinusoidal component of $f$ of frequency $n$; $r_n = \sqrt{a_n^2 +b_n^2}$ tells you its magnitude and $\phi_n$ tells you its phase.