The sine wave

Where does the sine wave come from? It's not a separate idea — it's the unit circle in disguise.

The unit circle, unrolled

Place a point P on the unit circle. As you increase the angle θ from 0° to 360° (0 to 2π radians), P travels around the circle. At every moment, P has a y-coordinate. That y-coordinate is sin θ.

Now take those y-coordinates and plot them against θ:

The dashed line shows the connection: the height of P on the circle at 60° equals the height of the wave at θ = π/3.

Reading the key values

Because sin θ = y-coordinate of the unit circle point:

The wave rises from 0, peaks at 1, returns to 0, dips to −1, and returns to 0. That's one period — the length of one complete cycle. For y = sin θ, the period is 2π (≈ 6.28 radians, or 360°).

The cosine wave

The cosine wave is the same idea but for the x-coordinate of P:

• cos 0° = 1 (rightmost)
• cos 90° = 0
• cos 180° = −1 (leftmost)

Cosine is just the sine wave shifted left by 90°. Same shape, different starting point.

Why this matters

Every repeating signal in physics, engineering, and music is built from sine and cosine waves. Understanding the unit-circle origin means you'll never need to memorize the wave shape — you can reconstruct it from a circle.