The graphs of sine and cosine, those familiar smooth, repeating wave shapes, can look like an entirely separate topic from the unit circle they're actually defined from, but they're not a separate idea at all — they're the exact same circular rotation, just unrolled onto a straight horizontal axis. Sine and cosine are each defined as the vertical or horizontal coordinate of a point travelling around the unit circle, and graphing that same coordinate against how far the point has rotated produces the smooth, repeating wave shape everyone recognises as the sine or cosine graph.
The graph's height at each point is just the circle's coordinate at that angle
Picture a point travelling steadily around the unit circle, starting at its rightmost position and moving counterclockwise. As that point travels, its vertical height above the horizontal axis rises to a maximum, falls back through zero, drops to a minimum, and rises back to zero again, completing one full smooth cycle for each full trip the point makes around the circle. Plotting that same vertical height against the angle the point has rotated through, rather than against its position on the circle itself, produces exactly the sine graph — the wave's shape isn't an arbitrary or separately invented pattern; it's a direct, point-by-point unrolling of the circle's own vertical coordinate as the angle steadily increases.
Cosine traces the same relationship using the circle's horizontal coordinate instead
Cosine works through exactly the same underlying logic, just tracking the travelling point's horizontal coordinate instead of its vertical one, which produces a wave shape identical to sine's in every respect except for where it starts — cosine begins at its maximum height, since a point starting at the circle's rightmost position begins with its horizontal coordinate already at its own maximum value, while sine begins at zero, since that same starting point's vertical coordinate starts at zero. Because both graphs derive from precisely the same underlying circular rotation, just tracking a different one of the circle's two coordinates, sine and cosine's graphs are simply the same repeating wave shape, shifted relative to each other by a fixed, unchanging amount corresponding to a quarter of one full circular rotation.
What we're still unsure about
The direct mathematical relationship between the unit circle and the sine and cosine graphs is precisely defined, settled mathematics, not a matter of any genuine dispute. What's more a matter of pedagogy than mathematics is which order actually helps a given student build the clearest intuitive understanding — some maths teachers introduce the wave-shaped graphs first, since they're more immediately visually familiar, and only connect them back to the underlying unit circle afterward, while others start from the circle itself and derive the graph from it directly, the order this post follows — a genuine, ongoing pedagogical question about sequencing and clarity, not a disagreement over what the underlying mathematics itself actually says.
This sits inside Graphs of Trigonometric Functions, one of seven topics in Trigonometry, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.