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cos(k omega t) = 1/2 * e^(i k omega t) + 1/2 * e^(-i k omega t)

The trigonometric spectrum of cos(k omega t) is single amplitude of the cosine function at a single frequency of k on the real axis which is using the basis function of cosine, right?

The complex exponential spectrum of cos(k omega t) has two amplitudes at 1/2, one at k and -k.

I am confused what this x-axis is representing, i get its the called the frequency domain but it is infact the index k multiplied by omega where omega is fundamental frequency which is constant. And also its not technically the frequency domain because negative frequencies don't exist, but its commonly called freq domain.

I also get this x-y plane shows amplitudes but amplitudes of what? Its not amplitudes of a cosine because these exponentials are actually made up of sines and cosines by eulers formula which are orthogonal and can't be on the same axis. So this x-axis is of +- infinity of k times fundamental frequency but what exactly is this and its size.

Are the exponentials of cosine, which have opposite signs, basis functions that completely represent the space and they together show the rotational direction that can be represented by a cosine?

Im all muddled up.

Appreciate your advice, thanks