MATLAB - HELP IS NEEDED! PLEASE - THE QUESTION HAS ALL THE INFORMATION!!!!! PLEASE POST A CORRECT SOLUTION

(a) Write a function called trigseries which takes as inputs x (a vector of x-values), n (number of terms in the series), and options (a text string which can be sine, cosine or both). The second two inputs should be optional (with default values n=10, options='both', as appropriate). trigseries should have two outputs: the computed value of the appropriate trigonometric function(s) and the l₂ error norm compared with matlab's built-in trigonometric functions (use the command norm).

trigseries should use two subfunctions sine_series and cosine_series which approximate the value of the associated functions at x using the appropriate Taylor series centred around x=0.

(b) Test trigseries using an input x of 10 equally spaced values between 0 and pi/2. Plot the output for both sine and cosine compared with the built-in function values. Test that your function works properly with different numbers of input variables.

(c) Make a log-log plot of the absolute value of the error as a function of the number of terms in the Taylor series for sin(x) for n=1,...,120 terms for x=pi/2 and x=4pi/2. Why does the error eventually become constant? Why do the two values of x give different minimum errors? Explain the difference in terms of the Taylor remainder theorem and loss of floating-point precision by plotting the magnitude n_th term in the series for n=1,...,120 terms.