# Plotting the spectrum of a continuous sinusoid function

I am just learning about spectrums and I am a little confused. I have given a simple example and where I get stuck.

Plot the spectrum of $x(t) = 5\cos\left(800\pi t + \frac{3 \pi}{7}\right)$.

So I use reverse-Euler's to write this as $$x(t) = \frac{5 e^{i \frac{3 \pi}{7}}}{2} e^{i 800\pi t} + \frac{5 e^{-i \frac{3 \pi}{7}}}{2} e^{-i 800\pi t}$$

• What is the spectral amplitude in each case? I am unclear if the frequency of $800 \pi t$ has amplitude $5$ or $\frac{5 e^{-i \frac{3 \pi}{7}}}{2}$.

• If it is the second do I make two seperate plots for the complex and real parts, or do I just plot $\bigg\lvert \frac{5 e^{-i \frac{3 \pi}{7}}}{2}\bigg\rvert$?

Given your signal $x(t) = 5\cos(800\pi t + \frac{3 \pi}{7})$. And considering your correct decomposition of it into complex exponentials:
$$x(t) = \frac{5 e^{i \frac{3 \pi}{7}}}{2} e^{i 800\pi t} + \frac{5 e^{-i \frac{3 \pi}{7}}}{2} e^{-i 800\pi t}$$
$$X(\Omega) = 5\pi \left( e^{i \frac{3 \pi}{7}} \delta(\Omega - 800\pi) + e^{-i \frac{3 \pi}{7}} \delta(\Omega + 800\pi) \right)$$
Now your Fourier transform has two impulses with weights of $5\pi e^{i \frac{3 \pi}{7}}$ and $5\pi e^{-i \frac{3 \pi}{7}}$ respectively.