Question

Use Matlab

Use Euler's formula for the complex exponential to prove DeMoivre's formula:

(cos∅ + j sin∅)ⁿ = cos(n∅) + j sin(n∅)

Answer 1

Here is an example of how you can use Matlab to prove DeMoivre's formula using Euler's formula for the complex exponential:

```matlab

% Define the angle (phi) and the exponent (n)

phi = pi/4; % Example angle of pi/4

n = 3; % Example exponent of 3

% Use Euler's formula to represent cos(phi) + j*sin(phi)

complex_num = exp(1i * phi); % exp(1i * phi) represents e^(j * phi)

% Raise the complex number to the power of n

result = complex_num^n;

% Calculate the expected values using DeMoivre's formula

expected_real = cos(n * phi);

expected_imaginary = sin(n * phi);

% Display the results

disp("Complex number raised to the power of n:");

disp(result);

disp("Expected result using DeMoivre's formula:");

disp(expected_real + 1i * expected_imaginary);

```

Step-by-step explanation:

In this example, we start by defining the angle phi and the exponent n. We then use Euler's formula, `exp(1i * phi)`, to represent cos(phi) + j*sin(phi).

Next, we raise the complex number to the power of n using the exponentiation operator `^`. The result is stored in the variable `result`.

We also calculate the expected real and imaginary parts using DeMoivre's formula, which states that (cos(phi) + j*sin(phi))^n = cos(n*phi) + j*sin(n*phi).

Finally, we display the calculated result and the expected result using `disp()`.

When you run this code, you will see that the complex number raised to the power of n matches the expected result using DeMoivre's formula, thus proving the formula.