Before calculators, anyone who needed a sine or cosine value reached for a printed table, because those functions were originally defined purely as ratios of a right triangle's sides — opposite over hypotenuse, adjacent over hypotenuse. That definition works cleanly for angles between 0 and 90 degrees, the only angles a right triangle actually has room for, and then it simply stops making sense. The unit circle definition rebuilds sine and cosine from a circle instead of a triangle, and in doing so frees them to handle any angle at all.
From a triangle's ratio to a point's coordinates
Draw a circle of radius one centred at the origin of a graph. For any angle, measured going counterclockwise from the positive x-axis, follow a ray out from the origin at that angle until it crosses the circle. The coordinates of that crossing point are, by definition, cosine of the angle and sine of the angle. For any angle between 0 and 90 degrees, this matches the old right-triangle ratios exactly — it's the same numbers, just relabelled as coordinates on a circle instead of side lengths in a triangle. The difference only shows up once the angle leaves that narrow range, because a circle, unlike a triangle, has no edge to run out of.
Why negative and "too big" angles suddenly make sense
On the unit circle, an angle of 200 degrees, or -45 degrees, isn't a category error the way it would be for a right triangle — it's simply a different point on the same circle, with perfectly well-defined coordinates and therefore a perfectly well-defined sine and cosine. Sweep the ray all the way around the circle and the point returns to where it started every 360 degrees, which is exactly where the repeating, wave-like shape of a sine or cosine graph actually comes from: it's not an arbitrary property someone assigned to the function, it's a direct consequence of tracing a point around something that has no start or end.
What we're still unsure about
The tidy European textbook story — ancient Greek chord tables leading directly to modern trigonometry — understates a longer, more distributed history. Indian mathematicians, notably Aryabhata around the fifth century CE, developed something much closer to the modern sine function and reasoned about it in circle-based terms centuries before it reached Europe. That work passed through Islamic mathematics, where scholars including al-Battani refined and extended it, before eventually influencing European trigonometry. Exactly how much specific credit belongs at each stage of that transmission, and how independently different traditions arrived at similar circle-based reasoning, is still genuinely debated among historians of mathematics — a more accurate and more interesting story than the single-origin version most classrooms default to.
This sits inside The Unit Circle, one of seven topics in Trigonometry, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.