What Is a Sine Wave? See y=sin x on a Live Board

A sine wave is the graph of the sine function — the set of points $(x,\sin x)$ drawn in the plane. On a window that spans more than one full turn, the picture is a smooth oscillation that repeats: up to a peak, down through zero, to a trough, and back.

The live board below freezes $y=\sin x$. Drag the angle along the top and read the height, instead of memorizing a sketch.

Where the wave comes from

On the unit circle, an angle $x$ (in radians) picks a point on the circle. The height of that point is $\sin x$. Plot that height against the angle on a horizontal axis, and the heights line up into the sine wave.

So the wave is sine, drawn as a graph. Every horizontal position $x$ meets the curve at height $\sin x$. Walk once around the circle and the heights have already drawn one full copy of the wave. The next walk draws the same copy again, shifted by one full turn.

How to read one period

For plain $y=\sin x$:

1. Zeros at $x=0$, $\pi$, $2\pi$ (and every integer multiple of $\pi$).
2. Maximum $1$ at $x=\pi /2$.
3. Minimum $-1$ at $x=3\pi /2$.
4. Period $2\pi$ — after one full turn the heights repeat.
5. Amplitude $1$ — the peak sits one unit above the midline $y=0$.

Those five facts are enough to sketch one hump and one trough. The board shows the same facts as a curve. Drag the angle and the point rides that curve. The fixed marks sit on the zeros and on the high and low points of that first period.

Worked example

Read $x=\pi /2$ on the board. The height is $1$, so the point is $(\pi /2,1)$ — the first peak. Move to $x=\pi$: the height is $0$, a zero crossing. Move to $x=3\pi /2$: the height is $-1$, the trough. From $x=0$ to $x=2\pi$ you have walked one period and returned to height $0$.

Two angles on the way up are worth reading before the peak. At $x=\pi /6$ the height is $1/2$, so the point is $(\pi /6,1/2)$. At $x=\pi /3$ the height is $\sqrt{3}/2$, about $0.866$, so the point is $(\pi /3,\sqrt{3}/2)$. Both sit on the same rising side, between the zero at $0$ and the peak at $\pi /2$. After the peak the heights come back down through those same values: $\sqrt{3}/2$ again near $x=2\pi /3$, and $1/2$ again near $x=5\pi /6$, before the zero at $\pi$.

A scaled wave $y=A\sin (Bx)$ keeps the same shape: $A$ stretches the height (amplitude), and $B$ compresses or stretches the period to $2\pi /|B|$. The handbook page on sine and cosine on a live board writes those formulas beside figures you can drag. This post stays with the plain wave $y=\sin x$ so the period and amplitude are visible before the parameters move.

Drag the angle, keep the rule

The angle sits on a slider above the wave, from $0$ to $2\pi$. Drag it. The point is $(x,\sin x)$ for that angle, so it has to stay on the curve, and the height you read is the value of sine. It starts at $\pi /2$, height $1$. Slide toward $\pi$ and the height falls toward $0$. Keep going and the point passes the trough at height $-1$, then climbs back to height $0$ at $2\pi$. The fixed marks stay put while the point travels. That is the check that the figure is a sine wave: one height per $x$, repeating every $2\pi$.

If you are still building the habit of plotting pairs $(x,f(x))$ before specializing to sine, the post on how to graph a function and the handbook page on real functions and their graphs cover that first skill. Degrees versus radians for the same angle live on degrees and radians on a live board.

GoodMaith keeps a relation like this as an object on a live board: name $y=\sin x$, drag the angle, and see why the repeating wave is exactly the list of heights from the unit circle. One full period forces every later copy.