Question

Problem 5 xx The Taylor series expansion for sin(x) is sin(x) = x -H + E-E+ = o E- (-1) . 2n +1 no (2n+1)! !57 where x is in radians. Write a MATLAB program that determines sin(x) using the Taylor series expansion. The program asks the user to type a value for an angle in degrees. Then the program uses a while loop for adding the terms of the Taylor series. If an n is the nth term in the series, then the sum Sne of the n terms is S, =Sau + E. In each pass calculate the estimated error E given by E = sner said. Stop adding terms when E 1E-8. The program displays the - Sold value of sin(s). Use the program for calculating sin(45) and sin(195) Show the results as: The sing of L is providing an error equal to L. Use scientific notation with 2 decimal places Problem 6 Write a MATLAB program for problem 5 using a for loop. Run the program for 5, 10, and 15 steps. Display the error at the end of the calculations.

Answer 1

Matlab Code:

Problem 5:

% MATLAB Program that calculates Taylor series expansion of sin(x)

clc;
clear;

% Reading angle in degrees
angleDegrees = input("Enter angle in degress: ");

% Converting angle in to radians
angleRadians = (0.0174533) * angleDegrees;

% Initially assign value 1
EstError = 1;

% Variable n for series
n = 0;

% Assign 0 to series sum
SNew = 0;
SOld = 0;

% Iterate till E > 1E-8
while EstError > (1e-8)

    % Calculating term
    term = ( ((-1)^n) / (factorial((2*n) + 1)) ) * ( angleRadians ^ ((2*n) + 1) );

    % Adding term
    if n == 0
      SOld = term; % Updating SOld
      EstError = abs((SNew - SOld)/SOld); % Calculating E
    else     
      SNew = SOld + term; % CAdding term to series
      EstError = abs((SNew - SOld)/SOld); % Calculating E
      SOld = SNew; % Updating SOld value
    end
  
    % Updating n value  
    n = n + 1;

end % End of while

% Printing resultant values
fprintf("\n The sin of %.2f is %.8f, providing an error equal to %.2e \n", angleDegrees, SOld, EstError);

Sample Run:

Enter angle in degress: 45 The sin of 45.00 is 0.70710702, providing an error equal to 2.49e-009 7

________________________________________________________________________________________________

Problem 6:

% MATLAB Program that calculates Taylor series expansion of sin(x)

clc;
clear;

% Reading angle in degrees
angleDegrees = input("Enter angle in degress: ");

% Converting angle in to radians
angleRadians = (0.0174533) * angleDegrees;

% Initially assign value 1
EstError = 1;

% Assign 0 to series sum
SNew = 0;
SOld = 0;

% Iterate for loop from 1 to 15
for n=0:14

    % Calculating term
    term = ( ((-1)^n) / (factorial((2*n) + 1)) ) * ( angleRadians ^ ((2*n) + 1) );

    % Adding term
    if n == 0
      SOld = term; % Updating SOld
      EstError = abs((SNew - SOld)/SOld); % Calculating E
    else     
      SNew = SOld + term; % CAdding term to series
      EstError = abs((SNew - SOld)/SOld); % Calculating E
      SOld = SNew; % Updating SOld value
    end
  
end % End of while

% Printing resultant values
fprintf("\n The sin of %.2f is %.8f, providing an error equal to %.2e \n", angleDegrees, SOld, EstError);

Sample Run:

_________________________________________________________________________________________

Problem 8:

function result = sinSeries_8(angleDegrees)
    #MATLAB function that reads angle in degrees and then calculates and returns value of sin(x)
  
    % Reading user choice
    ch = input("\n 1 - For Loop \t 2 - While Loop \t Your Choice: ");

    % Converting angle in to radians
    angleRadians = (0.0174533) * angleDegrees;

    % Initially assign value 1
    EstError = 1;

    % Variable n for series
    n = 0;

    % Assign 0 to series sum
    SNew = 0;
    SOld = 0;

    % Using for loop
    if ch == 1
      
        % Iterate for loop from 1 to 15
        for n=0:14
        
            % Calculating term
            term = ( ((-1)^n) / (factorial((2*n) + 1)) ) * ( angleRadians ^ ((2*n) + 1) );
        
            % Adding term
            if n == 0
              SOld = term; % Updating SOld
              EstError = abs((SNew - SOld)/SOld); % Calculating E
            else     
              SNew = SOld + term; % CAdding term to series
              EstError = abs((SNew - SOld)/SOld); % Calculating E
              SOld = SNew; % Updating SOld value
            end
          
        end % End of while

        % Printing resultant values
        fprintf("\n The sin of %.2f is %.8f using for loop, which provides an error equal to %.2e \n", angleDegrees, SOld, EstError);

      
    % Using while loop
    else
  
        % Iterate till E > 1E-8
        while EstError > (1e-8)
        
            % Calculating term
            term = ( ((-1)^n) / (factorial((2*n) + 1)) ) * ( angleRadians ^ ((2*n) + 1) );
        
            % Adding term
            if n == 0
              SOld = term; % Updating SOld
              EstError = abs((SNew - SOld)/SOld); % Calculating E
            else     
              SNew = SOld + term; % CAdding term to series
              EstError = abs((SNew - SOld)/SOld); % Calculating E
              SOld = SNew; % Updating SOld value
            end
          
            % Updating n value  
            n = n + 1;

        end % End of while
      
        % Printing resultant values
        fprintf("\n The sin of %.2f is %.8f using while loop, which provides an error equal to %.2e \n", angleDegrees, SOld, EstError);
      
    end
  
end

Sample Output:

Command Window >sinSeries_8 (45) 1 For Loop 2 While Loop The sin of 45.00 is 0.70710702 using for loop, which provides an error equal to 0.00e+000 Your Choice: 1 >sinSeries_8 (45) 1 For Loop2 While Loop Your Choice: 2 The sin of 45.00 is 0.70710702 using while loop, which provides an error equal to 2.49e-009