Take the sine of every angle — \sin 0^\circ, \sin 1^\circ, \sin 2^\circ, all the way round and round — and plot the answers. What you get is not a jumble of dots but a single, gorgeous, endlessly repeating curve: the sine wave.

This shape is nature's signature. A plucked guitar string, a ripple on a pond, the swing of a pendulum, the rise and fall of the tides, the alternating current in the wire behind your wall — all of them trace this exact curve. Learn to read it and you are reading the fundamental shape of sound, light, and vibration itself. So before we combine or transform it, let's get to know the plain curve and its personality.

Three numbers control its shape:

• Amplitude A — how tall the wave is. It stretches the curve vertically, so the peaks reach A and the troughs reach -A.
• Frequency f — how often it repeats. A larger f squeezes the wave horizontally, so more cycles fit across the same width.
• Phase \varphi — how far it is shifted sideways. A positive \varphi slides the whole wave to the left.

The shape of the plain wave

Start with the simplest case, y = \sin x (that is A = 1, f = 1, \varphi = 0). Every important property of the sine curve is already here:

• It oscillates between -1 and +1 — its amplitude is 1. It never escapes that band.
• It repeats every 360^\circ (that is 2\pi radians) — this repeat length is its period. After one full turn you are back exactly where you started.
• Its landmarks over one cycle: it passes through 0 at 0^\circ, climbs to a peak of +1 at 90^\circ, drops back to 0 at 180^\circ, sinks to a trough of -1 at 270^\circ, and returns to 0 at 360^\circ.
• It is full of symmetry: rotate it half a period about a zero and it maps onto itself (\sin(-x) = -\sin x, an odd function), and it repeats by simple sideways sliding. In fact cosine is the very same wave shifted 90^\circ to the left: \cos x = \sin(x + 90^\circ).

Play with the wave

Drag a slider (or type a value) and watch the curve respond. The angle x runs along the bottom in degrees. The faint line is the plain \sin(x) for comparison; the bold curve is y = A\,\sin(f x + \varphi). Push Amplitude up and watch the peaks grow taller; push Frequency up and watch the cycles bunch closer together.

Worked example 1 — reading a graph

A sine graph is drawn. Its peaks reach +3 and its troughs reach -3, and one full cycle takes 180^\circ before the pattern repeats. What is its equation, in the form y = A\,\sin(f x)?

• The peak height gives the amplitude directly: A = 3.
• The period is 180^\circ. Since a plain sine repeats every 360^\circ and here it repeats in half that, the wave is squeezed by a factor of 2: f = \dfrac{360^\circ}{\text{period}} = \dfrac{360}{180} = 2.

Worked example 2 — using symmetry to find every solution

Solve \sin\theta = 0.5 for 0^\circ \le \theta \le 360^\circ. Your calculator gives one answer, \theta = 30^\circ — but the curve's symmetry insists there is a second. The wave is symmetric about its peak at 90^\circ, so the value it hits on the way up at 30^\circ it hits again on the way down at the mirror-image angle:

So within one turn there are two solutions, 30^\circ and 150^\circ. Read only the first peak and you would have missed half the answer — an idea we push further in solving trig equations.

Worked example 3 — how y = a\sin(b\theta) changes things

The two numbers pull in opposite directions, which is the classic thing to keep straight:

• The amplitude is a, the number in front. Bigger a makes a taller wave. For y = 4\sin\theta, the amplitude is 4 (peaks at +4, troughs at -4).
• The period is \dfrac{360^\circ}{b}, set by the number multiplying \theta. Bigger b makes a shorter (more crowded) wave. For y = \sin(3\theta), the period is 120^\circ — three whole cycles squeezed into one normal turn.

These are exactly vertical and horizontal stretches — the same machinery as in transformations of graphs, just applied to a wave.