Where Did Sine Come From? The Strange History of a Simple Ratio — Part 1

Where did sine come from? We usually meet sine as a formula to memorise: sin θ = opposite / hypotenuse. But where did that ratio actually come from? And why is it called sine? Let's build it from scratch. We start with something much older: a triangle. From there, we discover why similar triangles create ratios that depend only on an angle, and how those ratios become sine, cosine and tangent. Then we leave the triangle behind and move into circles, chords and half-chords — following the geometry that astronomers used long before calculators existed. And eventually, we uncover the strange journey behind the word sine: from a Sanskrit word for a bowstring, through Arabic, and finally into Latin. No memorisation. Every idea is derived on screen. What you'll come away with: • Why similar triangles create constant side ratios • Why a ratio can stand in for an angle • Where sine, cosine and tangent actually come from • Why cosine is literally the sine of the complementary angle • How geometry gives us exact values for 30°, 45° and 60° • How circles and chords let us calculate angles we can't get from simple triangles • How the half-chord became the quantity we now call sine • The strange journey from jya → jiba → jayb → sinus • How Aryabhata's tradition built sine tables more than a thousand years ago • Why sine grows more slowly as the angle approaches 90° • Why sine was never really about triangles in the first place Chapters: [0:00] What exactly tells us what a triangle is? [1:45] Similar triangles and ratios [3:25] Why similar triangles work [4:55] Ratios become measurements of angles [6:42] Sine, cosine and tangent [8:18] Why cosine is called cosine [9:20] The problem with calculating sine [10:00] The angles we can get for free [12:23] From triangles to circles [12:54] Halving an angle [15:18] Building a table of angles [16:15] Where the word "sine" came from [17:32] Aryabhata's sine table [18:37] Why the numbers get smaller [19:14] Sine was never really about triangles [19:40] Part 2 This is Part 1 of a two-part series. In Part 2, we throw the triangle away completely and find out what sine really is. We'll build the connection between sine and cosine, learn how to add angles together, solve triangles without a right angle, and finally see how a spinning circle turns into a wave. Part 2 → [link] Animated with Manim, the open-source mathematical animation library. #math #trigonometry #sine #mathematics #education