Solution

Inserting the data below:

$$(t_0, y_0)=(0.25, 3)$$

$$(t_1, y_1)=(0.5, 2)$$

$$(t_2, y_2)=(0.75, -3)$$

$$(t_3, y_3)=(1, 0)$$

in the equation

$$y=a\space sin(\pi t)+b\space sin(2\pi t)+c\space sin(3\pi t)+d\space sin(4\pi t)$$

gives the following system of equations

$$\begin{bmatrix}\frac{1}{\sqrt{2}} & 1 & \frac{1}{\sqrt{2}} & 0 \\1 & 0 & -1 & 0 \\\frac{1}{\sqrt{2}} & -1 & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 0& 0\\ \end{bmatrix}\begin{bmatrix}a \\ b \\ c \\d \end{bmatrix}= \begin{bmatrix}3 \\ 2 \\ -3 \\0 \end{bmatrix}$$

Now let’s implement it in Python and solve it.