How Do You Add Two Sinusoidal Functions With the Same Frequency but Different Phase Shifts?

You add them by representing each sinusoid as a complex phasor, adding the phasors, and converting the result back into one sinusoidal function. The frequency remains unchanged, while the resulting amplitude and phase shift come from the modulus and argument of the phasor sum.

The Complex-Number Method

For

$$f(t)=A\cos(\omega t+\alpha)+B\cos(\omega t+\beta),$$

associate the terms with the phasors $Ae^{i\alpha}$ and $Be^{i\beta}$. Their sum is

$$C=Ae^{i\alpha}+Be^{i\beta}=Re^{i\phi},$$

where the resultant amplitude is $R=|C|$ and the resultant phase shift is $\phi=\arg(C)$. Therefore,

$$f(t)=R\cos(\omega t+\phi).$$

This works because multiplication by $e^{i\omega t}$ supplies the shared frequency, while the phasors encode the amplitudes and phase shifts.

To compute the sum in Cartesian form, resolve it into real and imaginary components:

$$x=A\cos\alpha+B\cos\beta,\qquad y=A\sin\alpha+B\sin\beta.$$

Then

$$R=\sqrt{x^2+y^2},\qquad \phi=\arg(x+iy).$$

An equivalent amplitude formula is

$$R=\sqrt{A^2+B^2+2AB\cos(\alpha-\beta)}.$$

This follows from the cosine rule for the angle between the two phasors. It also shows how interference works: if the phase difference is $0$, then $R=A+B$; if it is $\pi$, then $R=|A-B|$. For any other phase difference, the amplitude lies between these values. The phase cannot normally be found by averaging and , because the two phasors may have different amplitudes.

Use the two-argument inverse tangent on your GDC for $\phi$, since it selects the correct quadrant. A one-argument calculation can produce the wrong phase when $x<0$.

Worked Example

Simplify

$$3\cos(2t)+4\cos\left(2t+\frac{\pi}{2}\right).$$

The phasors are $3e^{i0}=3$ and $4e^{i\pi/2}=4i$, so $$. Hence,

$$R=\sqrt{3^2+4^2}=5, \qquad \phi=\tan^{-1}\left(\frac{4}{3}\right)\approx0.927.$$

Thus,

$$3\cos(2t)+4\cos\left(2t+\frac{\pi}{2}\right)=5\cos(2t+0.927).$$

Expanding the final expression confirms that its cosine and sine coefficients match the original sum.

A common misconception is to add amplitudes and phase shifts separately. This is incorrect because phase shifts represent directions, so the phasors must be added vectorially.

IB Exam Technique

This is AHL 1.13, Complex numbers continued, so it is HL-only content. Show the phasor sum before using technology, retain full precision for $\phi$, and convert the result back into a sinusoidal equation rather than leaving it in complex form.