Exploring Fourier Series: An=An*sin() & bn=An*cos()

[RaduAndrei]

Consider the following article:
https://en.wikipedia.org/wiki/Fourier_series

At definition, they say that an = An*sin() and bn = An*cos()

So with these notations you can go from a sum having sin and cos to a sum having only sin but with initial phases.

Why can I write an = An*sin() and bn = An*cos() ?
It seems out of the blue.

 

[blue_leaf77]

Substitute the 2nd equation to the first equation.

 

[RaduAndrei]

I know that by substitution we get from one form to another.
But my question is why I can write cos(phi) = a/sqrt(a^2+b^2) and sin(phi) = -b/sqrt(a^2+b^2) ?
I see that by taking cos(phi)^2 + sin(phi)^2 I get 1, so is good.

But why I can write cos(phi) like that? Writing cos(phi) like that, then from cos(phi)^2 + sin(phi)^2 = 1, I get sin(phi). But why I can write cos(phi) in the first place like that?

It is just arbitrary? If I write cos(phi) = a, then I find sin(phi)...then, ok. Is fine.I can see that. But writing as a/sqrt(a^2+b^2), it does not seem so straight-forward. Maybe there is a property that for any two numbers a,b then I can write cos(phi) in that way. I do not know.Going from the trigonometric Fourier sum to the exponential form, we use Euler's formula to write cos() = 1/2(e^+e^) and sin too. So I have Euler's formula here.

 

[blue_leaf77]

Expression like ##A_n \cos \phi_n## only depends on the index ##n##, so there is no harm in writing them in a more simple way such as ##a_n##.

Maybe there is a property that for any two numbers a,b then I can write cos(phi) in that way.

If you want to picture it that way, you first have to draw a right triangle and define which sides ##a## and ##b## correspond to, and which angle ##\phi## corresponds to.

 

[RaduAndrei]

Aa, ok. Now makes sense. Thanks.