Why does sin(x) make a wave?

When we plot y = sin(x), why does the graph come out as that endlessly repeating up-and-down wave, rather than some other shape?

Picture a point moving anticlockwise around a circle of radius 1, and keep your eye only on its height above the centre. As the point climbs the right side, the height rises to 1; as it crosses the top and comes down the left side, the height falls back through 0 down to −1; then it rises again.

The graph of sin(x) is nothing more than that height, unrolled along a straight line. The wave repeats because the point keeps going around the same circle — one full lap is one full wave.

Define sin(θ) as the y-coordinate of the point at angle θ on the unit circle. Because angles repeat every full turn, sin(θ + 2π) = sin(θ) — the function is periodic with period 2π.

The circle's radius bounds the height, so −1 ≤ sin(θ) ≤ 1: the wave's amplitude is 1. The height changes fastest where the point moves vertically (at θ = 0, π, …) and pauses at the top and bottom (θ = π/2, 3π/2) — which is why the wave is steepest at its zero crossings and flat at its peaks.

Take θ = π/6 (30°). The point on the unit circle at 30° has height 1/2, so sin(π/6) = 0.5. Half a turn later, at θ = π/6 + π, the point is diametrically opposite, at height −1/2 — the same value flipped, exactly what the wave shows one half-period along.

• Thinking the wave shape is a definition. It isn't — it's a consequence of circular motion. Change the circle's radius and you scale the amplitude; go around faster (sin(2x)) and you compress the period to π.
• Mixing degrees and radians when plotting: sin(x) in radians completes a wave every 2π ≈ 6.28, not every 360 units.

• NCERT Class 11 Mathematics, Ch. 3 — Trigonometric Functions
• Khan Academy — Unit circle definition of sin, cos, tan