A Familiar Claim, Put to the Test
While reviewing an electronics textbook, I came across a casual remark: sinusoidal signals of the same frequency always sum to another sinusoid, even when their magnitudes and phases differ. At first glance, this is not obvious. With different phases, shouldn't the signals interfere in a more complicated way? The claim deserves a closer look.
In engineering notation, a sinusoid with magnitude a, frequency ω, and phase φ is written as 
. The assertion is that the sum of two such signals with the same ω is also a sinusoid at that frequency, with a possibly different magnitude and phase.
Seeing Is Believing
Before diving into a formal proof, it helps to check whether the claim even holds. Using Python, Numpy, and matplotlib, I plotted two same-frequency sinusoids with different magnitudes and phases:
Adding them together and plotting the result in green:
The result looks like a clean sinusoid. Time to prove it.
A Trigonometric Proof
The first approach relies only on standard trigonometric identities. We start with the sum of two sinusoids and expand each term using the identity for the sine of a sum:
Applying this to both terms and regrouping gives an expression with sin(ωt) and cos(ωt) parts.
The next step is a change of variables. We assume we can find some 
and 
that satisfy a pair of equations derived from the regrouped terms. Squaring both equations and adding them eliminates the phase variable:
Using the Pythagorean identity, we obtain a direct formula for the magnitude in terms of the original amplitudes and phases. Dividing one equation by the other gives the tangent of the phase , and hence the phase itself:
With and determined, we substitute back into the sum's expression. Applying the sine-of-a-sum identity once more reveals that the sum is exactly a single sinusoid of the form:
The result confirms: the sum is a sinusoid with the same frequency , and we now have explicit formulas for its magnitude and phase in terms of the original parameters.
A More Elegant Complex Proof
For a cleaner derivation, we can work on the complex plane using Euler's formula. Instead of summing real sinusoids directly, we sum two complex exponentials:
Using Euler's identity, we factor out the common frequency term. The expression in brackets is a complex number in rectangular form. Converting it to polar form yields a magnitude and a phase. The magnitude follows from the Pythagorean combination of the real and imaginary parts, and the phase from their ratio.
The sum of the complex exponentials is therefore another complex exponential with the same frequency, but a new magnitude and phase:
Taking the imaginary part of this equation—via Euler's formula again—recovers the original statement for sines, with the previously computed and magnitude. Comparing real parts gives the analogous result for cosines, which is expected since a cosine is just a phase-shifted sine.
Both proofs show that the textbook's offhand comment is correct: same-frequency sinusoids, regardless of their individual magnitudes and phases, combine into a single sinusoid at that frequency. The key is that the sum's parameters are fully determined by the inputs, and the math works out cleanly whether you prefer the trigonometry or the complex exponentials.
