Question

The purpose of this question is to calculate the three cubic roots of a complex number. A complex number is of the form a + ib where i is v-1. The magnitude r of a complex number is Vab. The complex number a + ib can be written as r(cos θ + i sin θ). Therefore a -r cose and b rsin0 and b/a (r sin0)/(r cos0) - tane e- arctan(b/a). The 3 cubic roots of a complex number are computed using the following expression: + i sin There is one root for k 0, a second root for k 1 and a third root for k 2. In addition to finding the cubic roots of a complex number the program must calculate and display the value of the cube of each of the roots to verify that each root is correct. Use input to read in the values for a, b. The type of a and b must be float. The value entered for a must not be zero as that will cause a divide by zero error. When displaying the output be sure to use the correct format codes. See the next page for sample input to and output from the program. Your input and output should match the sample as closely as possible

Answer 1

% Matlab program to calculate the three cubic roots of a complex number

% input of a and b
a = input(' Enter the real part of the complex number : ');
while(a == 0) % check if a is 0
a = input(' a cannot be 0. Enter the real part of the complex number : ');
end
b = input(' Enter the imaginary part of the complex number : ');
% print the complex number
fprintf(' The complex number is (%g+%gj)',a,b);
theta = atan(b/a); % calculate theta
r = a/cos(theta); % calculate r
% calculate the roots
root1 = (r^(1/3))*((cos(theta/3))+(1i*sin(theta/3)));
root2 = (r^(1/3))*((cos((2*pi+theta)/3))+(1i*sin((2*pi+theta)/3)));
root3 = (r^(1/3))*((cos((4*pi+theta)/3))+(1i*sin((4*pi+theta)/3)));
% print the output
fprintf(' root1 = %f%+fj',real(root1),imag(root1));
fprintf(' root1^3 = %f%+fj',real(root1^3),imag(root1^3));
fprintf(' root2 = %f%+fj',real(root2),imag(root2));
fprintf(' root2^3 = %f%+fj',real(root2^3),imag(root2^3));
fprintf(' root3 = %f%+fj',real(root3),imag(root3));
fprintf(' root3^3 = %f%+fj ',real(root3^3),imag(root3^3));
%end of program

Output:

Enter the real part of the complex number 6.75 Enter the imaginary part of the complex number : 4.37 The complex number is (6.75+4.371) rootl-1.966790+0.381347j root!"3 = 6.750000+4·370000j root2-1.313652+1.512617j root2-3 = 6.750000+4.3700001 root3 = -0.653139-1.8939641 root3 36.750000+4.370000j