Sine Wave Graph: y = sin(x)
The sine curve is the shape of anything that repeats smoothly: sound, alternating current, tides, daylight hours through the year. It oscillates between −1 and 1 and repeats every 2π.
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What you are looking at
The teal curve is y = sin(x) and the rose curve is y = cos(x). They are the same shape, offset by a quarter of a cycle. Sine passes through the origin heading upward; cosine starts at its maximum.
The x-axis here is labelled in multiples of π rather than decimals, which is how trigonometric graphs are normally read. The π button in the calculator toolbar switches any graph to this mode.
Reading the wave
Four measurements describe any sine or cosine curve completely.
| Feature | For y = sin(x) | Meaning |
|---|---|---|
| Amplitude | 1 | Distance from the midline to a peak |
| Period | 2π ≈ 6.283 | Horizontal distance for one full cycle |
| Midline | y = 0 | The horizontal line the wave oscillates about |
| Range | [−1, 1] | Every output value it produces |
Where the shape comes from
Take a point moving at constant speed around a circle of radius 1. Its height above the centre, plotted against the angle turned, traces exactly y = sin(x). Its horizontal distance from the centre traces y = cos(x).
That is why the period is 2π: one full trip around a circle is 2π radians. It is also why sine and cosine are offset by a quarter turn — they measure two perpendicular directions of the same motion.
Why it matters outside mathematics
A pure musical tone is a sine wave, and its frequency is the pitch you hear. Mains electricity alternates as a sine wave. Any repeating signal, however complicated, can be built up from sine waves of different frequencies, which is the idea behind Fourier analysis and behind most audio and image compression.
Questions
Why is the x-axis in radians rather than degrees?
Radians make the mathematics cleaner: the derivative of sin(x) is cos(x) only when x is in radians. Degrees add a conversion factor to every calculus result. You can still enter degrees by typing the degree symbol after a number.
What is the difference between sine and cosine?
Only the starting point. cos(x) = sin(x + π/2), so cosine is sine shifted left by a quarter cycle. Either can describe any wave; you pick whichever gives the simpler phase shift.
Why does the sine curve never go above 1?
Because it is the height of a point on a circle of radius 1. The point cannot be further from the centre than the radius, so the output is trapped between −1 and 1.