Plotting the spectrum of a continuous sinusoid function
Given your signal $x(t) = 5\cos(800\pi t + \frac{3 \pi}{7})$. And considering your correct decomposition of it into complex exponentials:

$$x(t) = \frac{5 e^{i \frac{3 \pi}{7}}}{2} e^{i 800\pi t} + \frac{5 e^{-i \frac{3 \pi}{7}}}{2} e^{-i 800\pi t}$$

You next apply the continuous-Fourier transform properties to get:

$$X(\Omega) =  5\pi \left( e^{i \frac{3 \pi}{7}} \delta(\Omega - 800\pi) + e^{-i \frac{3 \pi}{7}} \delta(\Omega + 800\pi) \right)$$

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. 

The spectral amplitudes are infinite, by definition of the impulse function, but they have the weight respectively. And for the magnitude you would take the absolute value of the amplitudes, which is therefore infinite. Yet again you can represent the magnitudes of those impulses by their weights' absolute value.