The wave they make

Until now $\theta$ has meant “an angle.” Drop that. Treat $\sin$ as an ordinary function: hand it a real number, get a real number back. The input does not have to be an angle in any picture. $\sin(1)$, $\sin(2.5)$, $\sin(100)$ are all perfectly good outputs.

Mechanically nothing changed. To compute $\sin(\theta)$ you still walk a distance $\theta$ around the unit circle and read off the height. But now $\theta$ is just “how far along the input axis we are,” and we are going to plot the output.

As you drag the point, the $\sin\theta$ readout is the height. Imagine pulling that height out sideways and plotting it against $\theta$. That is the next step.

Walk the input $\theta$ steadily from $0$ upward and plot $\sin\theta$ as you go.

At $\theta = 0$ the height is $0$. It climbs to $1$ at $\theta = \pi/2$, falls back through $0$ at $\theta = \pi$, down to $-1$ at $\theta = 3\pi/2$, and returns to $0$ at $\theta = 2\pi$. Then it does the exact same thing again, forever, because going around the circle again retraces the same heights.

The result is a smooth, endlessly repeating wave. Its height never exceeds $1$ or drops below $-1$, and one full copy takes a horizontal distance of $2\pi$ before repeating.

Now plot $\cos\theta$, the $x$-coordinate, the same way. You get a wave of the identical shape: same height limits, same repeat distance $2\pi$.

The only difference is where it starts. Cosine begins at its peak: $\cos 0 = 1$. Sine begins at zero on the way up. Slide the cosine wave to the right by a quarter of a full cycle and it lands exactly on the sine wave.

There is only one wave shape in this lesson. Sine and cosine are two readings of it, a quarter-turn apart.

Every wave you will ever care about is a plain sine wave with three things adjusted. The general form is

and each letter is one independent knob:

• $A$, the amplitude, sets the height of the peaks.
• $k$, the frequency factor, sets how tightly the wave is packed horizontally.
• $\phi$, the phase, slides the whole wave left or right.

The three handles on the curve each grab one knob. Drag the top handle to stretch the peaks, the side handle to squeeze the cycles closer, the baseline handle to slide it. Watch the equation update.

This is the confusion the widget exists to kill.

Amplitude is vertical. $A\sin\theta$ has peaks at height $A$ instead of $1$. Pulling the peaks taller never changes how often the wave repeats.

Period is horizontal: the input distance for one full cycle. Plain $\sin\theta$ has period $2\pi$. In $\sin(k\theta)$, the inside finishes a full $2\pi$ when $\theta$ has only travelled $2\pi/k$, so

Multiply the input by $k$ and the wave speeds up by a factor of $k$, so its period shrinks. Bigger $k$, tighter wave. They are separate handles in the widget because they are separate ideas.

Period is “input per cycle.” Its reciprocal is frequency: “cycles per unit input.” They move in opposite directions: squeeze the period and the frequency rises.

Engineers usually write a wave as $\sin(2\pi f\theta)$, where $f$ is the frequency directly. Check it: that has $k = 2\pi f$, so its period is $2\pi / (2\pi f) = 1/f$. A wave at $f = 60$ cycles per second repeats every $1/60$ of a second. The widget shows both numbers at once so you can watch them trade off.

The last knob, $\phi$, does not change the shape or the spacing at all. It slides the wave bodily along the input axis. A positive $\phi$ shifts it left, meaning the wave reaches each value earlier; a negative $\phi$ delays it.

This is exactly the relationship from earlier: $\cos\theta = \sin(\theta + \pi/2)$ is a sine wave with $\phi = \pi/2$. Toggle the cosine overlay in the widget and you see two waves of identical shape, offset by that one quarter-cycle of phase.

You now have the full vocabulary of a wave: amplitude, frequency, period, phase.

Hold onto the frequency knob in particular. A transformer has no built-in sense of where a word sits in a sentence. To give it one, it tags each position with a stack of these waves, every one ticking at a different frequency $k$. Sample them all at the same input and you get a list of numbers that is unique to that position. You will build that list by hand in lesson five, and the lesson before it shows what those waves do when you start rotating with them.