Unit Circle to Sine Wave

Overview

A brief visual demonstration of how the unit circle generates the sine function. Viewers see a point travel around the circle while its vertical projection draws the familiar sine wave, reinforcing the geometric definition of $\sin\ heta$.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Title "Unit Circle → Sine" fades in at the top. The left side shows an empty coordinate plane, the right side shows an empty graph axis.
2	Draw Unit Circle	~4s	A unit circle (radius 1) is drawn centered at the origin in the left area. Axes appear with tick marks.
3	Rotating Point & Projection	~12s	A point starts at $(1,0)$ and moves counter‑clockwise at a constant angular speed. A vertical line from the point to the x‑axis is shown, and the point’s y‑coordinate is plotted in real time on the right‑hand sine graph, tracing the wave as the angle increases from $0$ to $2\pi$.
4	Highlight Relationship	~6s	When the point reaches key angles ($0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$), the corresponding peaks/troughs on the sine curve are highlighted and a brief label $\sin\ heta = y$ appears near the projection line.
5	Outro	~3s	The sine curve remains, the rotating point fades out, and a final caption "The sine of an angle equals the y‑coordinate on the unit circle" appears at the bottom before everything fades.

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Layout

┌───────────────────────────────────────┐
│               TOP AREA                 │
├───────────────────────┬───────────────┤
│        LEFT AREA       │   RIGHT AREA  │
│  (Unit circle + axes)  │ (Sine graph)  │
├───────────────────────┴───────────────┤
│               BOTTOM AREA              │
└───────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title "Unit Circle → Sine"	Fades in at phase 1, stays visible until phase 5
Left	Coordinate plane with unit circle, rotating point, vertical projection line	Primary visual; axes appear in phase 2
Right	Empty axes that become the sine curve as the point rotates	Curve is drawn in phase 3; highlights added in phase 4
Bottom	Final caption "The sine of an angle equals the y‑coordinate on the unit circle"	Appears in phase 5, small font, fades out at end

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Notes

• All motions are smooth and use a constant angular speed so that one full rotation corresponds to one full period of the sine wave.
• Colors: unit circle in teal, rotating point in red, projection line in gray, sine curve in blue.
• No textual labels are used except for the title and final caption; angle values are shown only at the key‑angle highlights.
• Total runtime ≈ 28 seconds, well within the 30‑second guideline.