Fourier transformation and amplitude and phase.

In Harmonic tremors and spectrograms I claimed that the Fourier transformation can be used for frequency analysis, which is of course true, but I feel that I should prove the following lemma which proves it.

Lemma. a sin(x) + b cos(x) = (a²+b²)^½ sin(x+arccos(a/(a²+b²)^½) = (a²+b²)^½ sin(x+arcsin(b/(a²+b²)^½).

Proof. There is exactly one solution for the following set of equations

• f(0) = b,
• f'(0) = a,
• f''(x) = -f(x),

namely the left hand side of the equation above. But s sin(x+u) also fulfils the third condition, so in order to fulfil the first two, we must solve

• b = s sin(u) = s (1-cos(u)²)^½,
• a = s cos(u) = s (1-sin(u)²)^½

and going after s first we arrive at

• sin(u) = b/s, cos(u) = a/s,
• a = s (1-(b/s)²)^½ = (s²-b²)^½, b = s (1-(a/s)²)^½ = (s²-a²)^½

and squaring that we reach both ways

•  s² = a²+b².

Inserting the (positive) square root of this into the equations under the third last point, we only have to invert sine and cosine simultaneously, which can be done, since ( a/(a²+b²)^½, b/(a²+b²)^½ ) lies on the unit circle, to reach the result.