Properties of Complex Numbers (phasor notation)

[jeff1evesque]

1. Statement:
The Real Part of a "Complex Number is expressed as the following:
$$Real(A) = \frac{1}{2}(A + A*) = \frac{1}{2}(|A|e^{j\alpha} + |A|e^{-j\alpha}) = \frac{1}{2}|A|(2cos(\alpha)) = |A|cos(\alpha)$$. (#1)

The Imaginary Part of a "Complex Number" is expressed as the following:
$$Imag(A) = \frac{1}{2}(A - A*) = \frac{1}{2}(|A|e^{j\alpha} - |A|e^{-j\alpha}) = \frac{1}{2}|A|(2jsin(\alpha)) = j|A|sin(\alpha)$$. (#2)


2. Questions:
I was just curious how $$\frac{1}{2}|A|(2cos(\alpha))$$ was derived in equation (#1), and how $$\frac{1}{2}|A|(2jsin(\alpha))$$ was derived in equation (#2)?

thanks,


Jeff

 

[tiny-tim]

I was just curious how $$\frac{1}{2}|A|(2cos(\alpha))$$ was derived in equation (#1), and how $$\frac{1}{2}|A|(2jsin(\alpha))$$ was derived in equation (#2)?

e^jα = cosα + jsinα
e^-jα = cosα - jsinα

so e^jα + e^-jα = 2cosα
e^jα - e^-jα = 2jsinα