Use Matlab

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【General guidance】The answer provided below has been developed in a clear step by step manner.Step1/2Here's the MATLAB code for parts (a) and (b) separately:Part (a)% Given observationst = [0,30,60,90,120,150,180,210,240,270,300,330];y = [408,89,-66,10,338,807,1238,1511,1583,1462,1183,804];% Construct matrix An = length(t); % Number of observationsA = zeros(n, 12); % Initialize matrix AA(:, 1) = 1; % Set first column of A to 1for k = 1:5 A(:, 2*k) = cos(2*pi*k*t/360); A(:, 2*k+1) = sin(2*pi*k*t/360);endA(:, 12) = cos(12*pi*t/360);% Solve the linear system Ax = yx = A \ y';% Display the solutiondisp('Coefficients:');disp(x'); The code above corresponds to part (a) of the problem, where we need to set up and solve the linear system \( \mathrm{{A}{x}={y}} \) to find the coefficients \( \mathrm{{a}_{{0}},{a}_{{1}},\ldots,{a}_{{6}}} \) and \( \mathrm{{b}_{{1}},\ldots,{b}_{{5}}} \) that interpolate the given observations. Here's an explanation of the code:We first define the given observations as two vectors, t and y.We then initialize the matrix A with zeros, and set the first column to 1 (for the constant term in the trigonometric function).We then loop over the indices k from 1 to 5, and compute the columns of A corresponding to the cosine and sine terms of the trigonometric function. We use the cos and sin functions, along with the values in the t vector, to compute these columns.Finally, we compute the column of A corresponding to the cosine term with frequency 12, and solve the linear system using the backslash operator in Matlab (x = A \ y'). The resulting vector x contains the coefficients of the trigonometric function that interpolate the observ ... See the full answer