Least Squares Approx. If I was using a least squares approximation of the form $y = A_1 + A_2\sin(wx) + A_3\cos(wx)$, would you be minimising the function $\sum_{i=0}^n (y_i - (A_1 + A_2\sin(wx) + A_3\cos(wx))^2$ ?
I've never tried this for periodic data before!
 A: A straightforward method (no initial gess neeeded, no iterative computation) is shown pages 34-36 in :
http://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales
A: The least squares method takes a trial function, $y(x)$, and a set of $m$ data points $\left\{ x_{k}, y_{k} \right\}_{k=1}^{m}$ and provides the best solution vector $A=\left(A_1, A_2, A_3\right).$
Define the residual error as the difference between the data and the prediction as $r_k (A) = y_k - y(x_k).$ The method is named for minimizing the sum of the squares of the residual errors. That is, the solution vector is defined as
$$
  r^{2}(A) = \left\{ A \in \mathbb{R}^{3} \colon \sum_{k=1}^{m} \left(y_k - y(x_k) \right)^{2} \text{ is minimized} \right\}
$$
The system your problem is
$$
\left[
\begin{array}{ccc}
  1 & \sin (w x_1 ) & \cos (w x_1 ) \\
  1 & \sin (w x_2 ) & \cos (w x_2 ) \\
  \vdots & \vdots & \vdots \\
  1 & \sin (w x_m ) & \cos (w x_m )
\end{array}
\right]
\left[
\begin{array}{c}
  A_1 \\
  A_2 \\
  A_3
\end{array}
\right]
=
\left[
\begin{array}{}
  y_1  \\
  y_2   \\
  \vdots \\
  y_m  
\end{array}
\right],
$$
which is manifestly linear in the fit parameters. Solve via the normal equations if the data is well conditioned, or QR, SVD decompositions, etc.
So yes, you would be minimizing
$$
\begin{align}
  r^{2}(A_1,A_2,A_3) &= 
\Bigg\lVert
\left[
\begin{array}{ccc}
  1 & \sin (w x_1 ) & \cos (w x_1 ) \\
  1 & \sin (w x_2 ) & \cos (w x_2 ) \\
  \vdots & \vdots & \vdots \\
  1 & \sin (w x_m ) & \cos (w x_m )
\end{array}
\right]
\left[
\begin{array}{c}
  A_1 \\
  A_2 \\
  A_3
\end{array}
\right]
-
\left[
\begin{array}{}
  y_1  \\
  y_2   \\
  \vdots \\
  y_m  
\end{array}
\right]
\Bigg\rVert_{2}^{2} \\
&=
\sum_{k=1}^{m} \left( y_k - A_1 - A_2 \sin \left( w x_k \right) - A_3 \cos \left( w_k \right) \right)^2.
\end{align}
$$
Keep in mind that the best fit may not be a good fit. If you have a bad trial function, you will get the best fit for a bad model.
