Q:

Trigonometrical Least Squares (Linear Algebra)

Solutions which use matrices would be greatly appreciated.
Find the trigonometric function of the form $f(t) = c_0+c_1\sin(t)+c_2\cos(t)$ that best fits the data points $(0,0), (1,1), (2,2), (3,3)$ using least squares.
My question is, would I go about this in the same way as I would if I was using a least squares approach to fit 4 sets of coordinates to a quadratic line? Or does the trigonometry result in me having to go about this question in a different way? If so, how?

A:

You have a trial function and let's generalize the data to the sequence $\left\{ t_{k}, f_{k} \right\}_{k=1}^{m}$. The linear system takes the form
$$
\begin{align}
  \mathbf{A} \alpha &= f \\[3pt]
%
\left[ \begin{array}{ccc}
  \mathbf{1} & s & c
\end{array} \right] 
%
\left[ \begin{array}{c} 
  \alpha
\end{array} \right]
%
&=
\left[ \begin{array}{c}
 f
\end{array} \right]
%
\\[5pt]
%
\left[
\begin{array}{ccc}
  1 & \sin t_{1} & \cos t_{1} \\
  1 & \sin t_{2} & \cos t_{2} \\
  \vdots & \vdots & \vdots \\
  1 & \sin t_{m} & \cos t_{m}
\end{array}
\right]
%
\left[
\begin{array}{c}
  \alpha_{0} \\ \alpha_{1} \\ \alpha_{2}
\end{array}
\right]
%
 &=
%
\left[
\begin{array}{c}
 f_{1} \\ f_{2} \\ \vdots \\ f_{m}
\end{array}
\right].
\end{align}
$$
Set up the normal equations:
$$
\begin{align}
  \mathbf{A}^{*} \mathbf{A} \alpha &= \mathbf{A}^{*} f \\[3pt]
%
\left[
\begin{array}{ccc}
  \mathbf{1} \cdot \mathbf{1} &  \mathbf{1} \cdot s &  \mathbf{1} \cdot c \\
  s \cdot \mathbf{1} & s \cdot s & s \cdot c \\
  c \cdot \mathbf{1} & c \cdot s & c \cdot c
\end{array}
\right]
%
\left[
\begin{array}{c}
  \alpha_{0} \\ \alpha_{1} \\ \alpha_{2}
\end{array}
\right]
%
&=
%
\left[
\begin{array}{c}
  \mathbf{1} \cdot f \\ s \cdot f \\ c \cdot f
\end{array}
\right].
&
\end{align}
$$
Solve the normal equations:
$$
\begin{align}
   \alpha &= \left( \mathbf{A}^{*} \mathbf{A} \right)^{-1} \mathbf{A}^{*} f \\[3pt]
%
\left[
\begin{array}{c}
  \alpha_{0} \\ \alpha_{1} \\ \alpha_{2}
\end{array}
\right]
%
&=
%
\left[
\begin{array}{ccc}
  \mathbf{1} \cdot \mathbf{1} &  \mathbf{1} \cdot s &  \mathbf{1} \cdot c \\
  s \cdot \mathbf{1} & s \cdot s & s \cdot c \\
  c \cdot \mathbf{1} & c \cdot s & c \cdot c
\end{array}
\right]^{-1}
%
\left[
\begin{array}{c}
  \mathbf{1} \cdot f \\ s \cdot f \\ c \cdot f
\end{array}
\right].
&
\end{align}
$$

