Speaking as an EE, imaginary exponentials are generally easier to manipulate then their trigonometric counterparts since they essentially let you get a polynomial out of your sin/cos function.
For example if I gave you $ \cos \Bigg(n_1\omega t+\theta_1\Bigg)\cos\Bigg(n_2\omega t+\theta_2\Bigg) $
You can imagine trying to do any kind of algebra with that. Using exponentials we can write it in a simpler form.
$ \Bigg[\mathrm e^{\theta_1}\mathrm e^{jn_1\omega t}+\mathrm e^{\theta_1}\mathrm e^{-jn_1\omega t} \Bigg] \Bigg[\mathrm e^{\theta_2}\mathrm e^{jn_2\omega t}+\mathrm e^{\theta_2}\mathrm e^{-jn_2\omega t} \Bigg] $
Do some math and you get him down to $ \Bigg[\mathrm e^{\theta_1\theta_2}\mathrm e^{j\omega t(n_1+n_2)}+\mathrm e^{\theta_1\theta_2}\mathrm e^{-j\omega t(n_1+n_2)} \Bigg]+\Bigg[\mathrm e^{\theta_1\theta_2}\mathrm e^{j\omega t(n_2-n_1)}+\mathrm e^{\theta_1\theta_2}\mathrm e^{-j\omega t(n_2-n_1)} \Bigg] $
And simplify to $ \cos \Bigg[(n_1+n_2)\omega t + \theta_1\theta_2) \Bigg]+\cos \Bigg[(n_2-n_1)\omega t + \theta_1\theta_2) \Bigg] $
That looks much nicer doesn't it?