Why the polar form(2.83,45) is written a rectangular complex number like this:
2.83*cos45 + j*(2.83*sin45)
I would like some explication please.
Thanks!
The "j" means there is a 90-degree phase
difference between the two terms and they cannot be added directly.
Another method of writing the polar form: 2.83[45o].
The i and j are unit vectors in the x any y directions.
This vector is of magnitude 2.83 and its direction is 45 degrees counterclockwise from the x axis so its components are
X direction: 2.83 * cos 45
Y direction: 2/83 * sin 45
so:
i * (2.83*cos45) + j*(2.83*sin45)
The polar form of a complex number represents the number in terms of its magnitude (denoted by r) and angle (denoted by θ). In this case, the polar form (2.83, 45) means that the complex number has a magnitude of 2.83 and an angle of 45 degrees.
To convert the polar form to a rectangular form (also known as Cartesian form), which is in the form a + jb, where a and b are real numbers, we can use trigonometry.
In the rectangular form, the real part is the product of the magnitude and the cosine of the angle, and the imaginary part is the product of the magnitude and the sine of the angle.
So, for the given polar form (2.83, 45), we can write it as a rectangular complex number using the following formulas:
Real part = magnitude * cos(angle)
Imaginary part = magnitude * sin(angle)
Therefore:
Real part = 2.83 * cos(45)
Imaginary part = 2.83 * sin(45)
Simplifying these expressions:
Real part = 2.83 * √(2)/2
= 1.99935
Imaginary part = 2.83 * √(2)/2
= 1.99935
Hence, the rectangular form of the complex number (2.83, 45) can be written as 1.99935 + j * 1.99935.