Question

3. Cosine Approximation Write a function m-file to calculate the approximated cosine function which can be evaluated by the following infinite series - equation (2): 1. x² x4 26 (-1)(n+1). 22(n-1) cos x = 1 - 3 + - 6! + + (2(n − 1))! Here are requirements for the function m-file. - Parameter lists (inputs): 2 value (in radian), and desired error percentagel. - Return value (outputs): approximated value, real error percentage, and number of iterations - Function name: 'Approx_Cos_yourEmailAccount Using a script m-file which will call the Approx_Cos_yourEmailAccount function, save outputs with following given inputs. - x = 7/12 and %error = 1%. Save it on EX1_11.dat. - X= 7/12 and %error = 0.1%. Save it on EX1_12.dat. x = 7/6 and %error = 1%. Save it on EX1_13.dat. - * = 7/6 and %error = 0.1%. Save it on EX1_14.dat. - x = 7/6 and %error = 0.01%. Save it on EX1_15.dat.

Answer 1

PLEASE REFER BELOW INSTRUCTIONS

1) Create Approx_Cos_yourEmailAccount.m and paste below code

function [approx_value,real_per_error,iterations] = Approx_Cos_yourEmailAccount(x,desired_per_error)
    approx_value = 0.0;
    real_per_error = 0.0;
    iterations = 0;
    sign = 1;
    orig_cos_val = cos(x);
    power = 0;
    while true
        approx_value = approx_value + ((sign * x^power) / factorial(power));
        iterations = iterations + 1;
        real_per_error = abs(approx_value-orig_cos_val) * 100;
        sign = -1 * sign;
        power = power + 2;
      
        if( real_per_error <= desired_per_error)
            break;
        end
    end
end

2) create test.m and paste below code

close all
clear all
clc

x = pi / 12;
desired_per_error = 1;

fileID = fopen('EX1_11.dat','w');
[approx_value,real_per_error,iterations] = Approx_Cos_yourEmailAccount(x,desired_per_error)
test=[x desired_per_error];
fprintf(fileID,'%f %f\n',test.');
test = [approx_value real_per_error iterations];
fprintf(fileID,'%f %f %f\n',test.');
fclose(fileID);

x = pi / 12;
desired_per_error =0.1;

fileID = fopen('EX1_12.dat','w');
[approx_value,real_per_error,iterations] = Approx_Cos_yourEmailAccount(x,desired_per_error)
test=[x desired_per_error];
fprintf(fileID,'%f %f\n',test.');
test = [approx_value real_per_error iterations];
fprintf(fileID,'%f %f %f\n',test.');
fclose(fileID);

x = pi / 6;
desired_per_error = 1;

fileID = fopen('EX1_13.dat','w');
[approx_value,real_per_error,iterations] = Approx_Cos_yourEmailAccount(x,desired_per_error)
test=[x desired_per_error];
fprintf(fileID,'%f %f\n',test.');
test = [approx_value real_per_error iterations];
fprintf(fileID,'%f %f %f\n',test.');
fclose(fileID);

x = pi / 6;
desired_per_error = 0.1;

fileID = fopen('EX1_14.dat','w');
[approx_value,real_per_error,iterations] = Approx_Cos_yourEmailAccount(x,desired_per_error)
test=[x desired_per_error];
fprintf(fileID,'%f %f\n',test.');
test = [approx_value real_per_error iterations];
fprintf(fileID,'%f %f %f\n',test.');
fclose(fileID);

x = pi / 6;
desired_per_error = 0.01;

fileID = fopen('EX1_15.dat','w');
[approx_value,real_per_error,iterations] = Approx_Cos_yourEmailAccount(x,desired_per_error)
test=[x desired_per_error];
fprintf(fileID,'%f %f\n',test.');
test = [approx_value real_per_error iterations];
fprintf(fileID,'%f %f %f\n',test.');
fclose(fileID);

output will be .dat files and below

approx_value =

    0.9657

real_per_error =

    0.0195

iterations =

     2

approx_value =

    0.9657

real_per_error =

    0.0195

iterations =

     2

approx_value =

    0.8629

real_per_error =

    0.3103

iterations =

     2

approx_value =

    0.8661

real_per_error =

    0.0028

iterations =

     3

approx_value =

    0.8661

real_per_error =

    0.0028

iterations =

     3

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