A real cosine can be seen as the sum of two opposite rotating complex comjugate vectors:
\( \frac{1}{2}e^{j2\pi ft} \) and \( \frac{1}{2}e^{-j2\pi ft} \).
Try it by yourself — in this example, \( f = 1\,\text{Hz} \).
0.150 s
0 s1 s
Comments
As you can see,
\( \cos(2\pi ft) = \frac{1}{2}e^{j2\pi ft} + \frac{1}{2}e^{-j2\pi ft} \).
The imaginary parts cancel, while the real parts add together.
Comments
As you can see, \( \cos(2\pi ft) = \frac{1}{2}e^{j2\pi ft} + \frac{1}{2}e^{-j2\pi ft} \). The imaginary parts cancel, while the real parts add together.