Relationship between real and imaginary part of a real-valued and causal system
I suppose you've learned about the Hilbert transform relationship between the real part and the imaginary part of the Fourier transform of a real-valued and causal signal. So you can obtain the real part from the imaginary part using this relationship. Since the imaginary part is just composed of sinusoids, the real part can be obtained very easily (note that the Hilbert transform of a sine is a negative cosine).