Graphs of sine and cosine

Follow the height of a point going round a circle and plot it against the angle: one lap draws one full wave, 2π long.

From circle to wave

In the previous unit the sine of an angle was the height of a point on the unit circle. Now make the angle the horizontal axis of a graph, measured in radians, and plot that height above it. At 0 the point is at (1, 0), height 0. It rises to height 1 at π/2, comes back to 0 at π, sinks to −1 at 3π/2 and returns to 0 at 2π, where it started. The curve you get is the sine wave. Keep going round and the same shape repeats.

Drag to turn. Scroll or pinch to zoom.

Press Play, or drag the point on the circle or along the graph. The green curve is drawn up to the current angle. The purple curve is y = A sin(Bx); change A and B to stretch it.

Period and range

OpenStax Precalculus describes the main features. The graph repeats after 2π, so sine is a periodic function with period 2π: sin(x + 2π) = sin x for every x. The values never go above 1 or below −1, because the point stays on a circle of radius 1. The wave crosses the axis at every multiple of π, peaks at π/2 plus whole turns and bottoms out at 3π/2 plus whole turns.

Cosine is the horizontal coordinate, so its graph starts at 1 when x = 0, drops to 0 at π/2 and reaches −1 at π. It is the same wave as sine, slid a quarter turn to the left: cos x = sin(x + π/2). Turn on the cosine graph in the lab to see the two curves side by side, always a quarter period apart.

One period of sin x (solid) and cos x (dashed).

Stretching the wave

Precalculus writes the general sinusoidal function as y = A sin(Bx − C) + D. Here only A and B matter. The number A stretches the wave vertically; its absolute value |A| is the amplitude, the height of a peak above the middle line. So y = 3 sin x swings between 3 and −3. The number B squeezes the wave horizontally: the period becomes P = 2π/|B|. With B = 2 the wave repeats every π, twice as often; with B = 1/2 it takes 4π, about 12.57, to repeat. A negative A flips the wave upside down without changing its amplitude.

BackgroundThe name sine has an odd history, which MacTutor sets out. Aryabhata, in India around the year 500, made tables of half chords, which are true sine tables, and called the sine jya. Arab mathematicians took the word over as jiba, a sound with no meaning in Arabic. Later writers spelled it jaib, an Arabic word meaning a fold, and European translators rendered that into Latin as sinus, which also means a fold. Fibonacci's use of sinus rectus arcus helped it spread. The short form sin was first used by Edmund Gunter in 1624, in a drawing, and first appeared in a book in 1634, by the French mathematician Hérigone.

about 140 BC: Hipparchus makes the first known table of chords.
about 500: Aryabhata tabulates half chords and calls the sine jya.
628: Brahmagupta reproduces the same sine table.
980: Abu'l-Wafa knows sin 2x = 2 sin x cos x.
1533: Regiomontanus' De triangulis, written about 1464, is published.
1583: Thomas Fincke first uses the name tangent.
1624: Edmund Gunter first writes the abbreviation sin.

Period of y = sin x, to 4 decimal places?
Largest value of sin x?
At which x between 0 and 2π does sin x reach −1?
- 3π/2
- π
- π/2
- 2π
cos 0?
Amplitude of y = 3 sin x?
Amplitude of y = −2 sin x?
Period of y = sin 2x, to 4 decimal places?
Period of y = sin(x/2), to 2 decimal places?
cos x equals:
- sin(x + π/2)
- sin(x + π)
- −sin x
- sin 2x
The Latin word sinus that gave us sine means:
- a fold
- a chord
- a circle
- a shadow
Who first used the abbreviation sin, and in what year? Give the year.

Key points

sin x plots the height of a point on the unit circle against the angle.
Period 2π; values between −1 and 1; zeros at multiples of π.
cos x = sin(x + π/2): the same wave a quarter turn ahead.
y = A sin(Bx): amplitude |A|, period 2π/|B|.
jya → jiba → jaib → sinus; sin first written by Gunter in 1624.