Note that

\(\displaystyle{S}={R}{e}{\left({a}{e}^{{\alpha+\theta}}+{b}{e}^{{\beta+\theta}}\right)}\)

You can imagine these two complex numbers as vectors in the Argand plane.The resultant of these two vectors, can be found from the rule of addition of vectors and would have magnitude equal to \(\displaystyle\sqrt{{{a}^{{2}}+{b}^{{2}}+{2}{a}{b}{\cos{{\left(\alpha-\beta\right)}}}}}\). Now, the real part of this resultant vector would be S. This would be its cosine component, which lies between −1 and 1. Hence the result holds up.

\(\displaystyle{S}={R}{e}{\left({a}{e}^{{\alpha+\theta}}+{b}{e}^{{\beta+\theta}}\right)}\)

You can imagine these two complex numbers as vectors in the Argand plane.The resultant of these two vectors, can be found from the rule of addition of vectors and would have magnitude equal to \(\displaystyle\sqrt{{{a}^{{2}}+{b}^{{2}}+{2}{a}{b}{\cos{{\left(\alpha-\beta\right)}}}}}\). Now, the real part of this resultant vector would be S. This would be its cosine component, which lies between −1 and 1. Hence the result holds up.