By the end of this lesson, you should be able to:
- Identify amplitude, angular frequency, phase, period, and ordinary frequency in a sinusoidal function.
- Predict how changing A, ω, or φ changes the graph.
- Relate a rotating complex phasor to the cosine waveform through its real projection.
- Connect sinusoidal phase with the phase carried by complex quantum amplitudes.
The general sinusoidal function
A sinusoidal signal can be written as
\[x(t)=A\cos(\omega t+\phi).\]
The three parameters describe different properties of the oscillation. The amplitude \(A\) controls the maximum displacement from zero, the angular frequency \(\omega\) controls how rapidly the phase changes, and the phase \(\phi\) determines where in its cycle the oscillation begins.
Amplitude changes the vertical scale
For a cosine with no vertical offset, the signal ranges between \(-A\) and \(+A\). Increasing \(A\) makes peaks taller and troughs deeper, but it does not change the period or horizontal location of the waveform.
Use Amplitude mode in the laboratory and predict the new peak and trough before moving the slider.
Angular frequency controls the rate of oscillation
Angular frequency is measured in radians per unit time. It is related to the period \(T\) and ordinary frequency \(f\) by
\[T=\frac{2\pi}{\omega},\qquad f=\frac{1}{T}=\frac{\omega}{2\pi}.\]
Increasing \(\omega\) compresses the waveform horizontally because each complete \(2\pi\)-radian cycle takes less time.
Phase determines the starting point
The phase \(\phi\) is an angular offset. For \(x(t)=A\cos(\omega t+\phi)\), the corresponding horizontal time shift is
\[\Delta t=-\frac{\phi}{\omega}.\]
A positive phase therefore shifts this form of the cosine to the left. In Phase mode, a faint reference curve with \(\phi=0\) lets you compare the shifted waveform directly.
From circular motion to a sinusoid
Euler's formula connects complex rotation to sine and cosine:
\[e^{i\theta}=\cos\theta+i\sin\theta.\]
For a rotating phasor \(z(t)=Ae^{i(\omega t+\phi)}\), the real component is exactly
\[\operatorname{Re}\{z(t)\}=A\cos(\omega t+\phi).\]
Press Play in the interactive laboratory. The point moving along the sinusoid and the rotating phasor remain synchronized so you can see this relationship continuously.
Why this matters in quantum computing
Quantum amplitudes are complex. A complex amplitude can be written in polar form as \(re^{i\phi}\), where \(r\) is its magnitude and \(\phi\) is its phase. The same geometry you see in the rotating phasor appears when describing quantum states.
For example, a single-qubit state may contain a relative phase:
\[|\psi\rangle=\frac{|0\rangle+e^{i\phi}|1\rangle}{\sqrt2}.\]
The phase does not change the magnitudes of the two amplitudes, but it can change how they interfere after later quantum gates. This is why understanding sinusoidal phase and complex rotation provides a useful bridge to Bloch-sphere rotations and quantum interference.
Watch the phase, not only the height of the wave. In quantum computing, phase can carry information that becomes visible only after amplitudes interfere.
What should stay with you
- Amplitude A controls vertical extent.
- Angular frequency ω controls how rapidly the sinusoid oscillates, with T=2π/ω and f=ω/(2π).
- Phase φ shifts the oscillation horizontally without changing its amplitude or frequency.
- The cosine waveform is the real projection of the rotating complex phasor Ae^{i(ωt+φ)}.
If ω increases while A and φ stay fixed, what happens to x(t)=A cos(ωt+φ)?
Angular frequency controls how rapidly the phase angle changes. Increasing ω reduces the period T=2π/ω, so more cycles fit into the same time interval.