Question

Using Euler's formula, represent the following complex numbers in the Cartesian form (i.e., a+jb ).

(a) eʲπ/³
​(b) 2e⁻ʲπ/³

Answer 1

To represent complex numbers using Euler's formula, use the amplitude, wave number, and angular frequency. (a) The Cartesian form of e^(iπ/3) is (1/2) + (sqrt(3)/2)i. (b) The Cartesian form of 2e^(-iπ/3) is 1 - sqrt(3)i.

Step-by-step explanation:

To represent complex numbers in Cartesian form using Euler's formula, we can use the following steps:

(a) For the complex number e^(iπ/3):

- The amplitude (A) is 1.

- The wave number (k) is 0.

- The angular frequency (ω) is π/3.

Using Euler's formula, e^(iπ/3) can be written as: cos(π/3) + i sin(π/3).

Thus, the Cartesian form is: (1/2) + (sqrt(3)/2)i.

(b) For the complex number 2e^(-iπ/3):

- The amplitude (A) is 2.

- The wave number (k) is 0.

- The angular frequency (ω) is -π/3.

Using Euler's formula, 2e^(-iπ/3) can be written as: 2cos(-π/3) + 2i sin(-π/3).

Thus, the Cartesian form is: 1 - sqrt(3)i.