Sine and Cosine Unit Circle Explanation

Sine and Cosine from the Unit Circle

Overview

A 60‑second dark‑background animation that shows how the coordinates of a point rotating on the unit circle generate the sine and cosine waveforms. The key takeaway is that the horizontal coordinate equals $\cos\theta$ and the vertical coordinate equals $\sin\theta$.

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Phases

#	Phase Name	Duration	Description
1	Intro & Setup	~5s	Fade‑in dark background, draw axes for the unit circle on the left and the two empty graphs on the right (sin top, cos bottom). Label axes (x, y) and graph axes “Angle θ”.
2	Rotating Point Begins	~8s	Place a point at $(1,0)$ on the circle, show radius line and angle arc $\theta$. Start smooth counter‑clockwise rotation.
3	Projections Appear	~7s	As the point moves, draw orange horizontal projection to the x‑axis (label $x=\cos\theta$) and blue vertical projection to the y‑axis (label $y=\sin\theta$). Dashed lines keep the projections visible.
4	Graph Tracing – Sine	~12s	The blue vertical projection feeds a moving marker on the top‑right sine graph; the curve $y=\sin\theta$ is drawn progressively in sync with the rotation.
5	Graph Tracing – Cosine	~12s	Simultaneously, the orange horizontal projection feeds a marker on the bottom‑right cosine graph; the curve $y=\cos\theta$ is drawn in sync.
6	Key Angle Pauses	~12s	At $0°$, $90°$, $180°$, $270°$ and $360°$ pause ~2 s each. Show the exact sine and cosine values beside the point and on the graphs.
7	Summary & End	~4s	Fade‑in the final statement “Cosine gives horizontal position. Sine gives vertical position.” while keeping the circle and graphs visible for a brief moment.

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Layout

┌───────────────────────────────────────────────────────┐
│                     MAIN (dark background)           │
│                                                       │
│  ┌───────────────┐   ┌───────────────────────────────┐ │
│  │   Unit Circle │   │   Sine graph (top‑right)          │ │
│  │   (left)      │   │   (blue curve)                    │ │
│  └───────────────┘   └───────────────────────────────┘ │
│                           ┌───────────────────────┐ │
│                           │   Cosine graph (bottom)│ │
│                           │   (orange curve)       │ │
│                           └───────────────────────┘ │
│                                                       │
└───────────────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Unit circle with rotating point, radius line, angle arc, projection lines, and two synchronized graphs (sine top, cosine bottom). Large coloured labels for $x=\cos\theta$ and $y=\sin\theta$.	Dark background, high‑contrast orange and blue colors, axis ticks at –1, 0, +1.
Caption	Optional brief step titles (e.g., “Rotate”, “Project”, “Trace”).	Small font, placed at bottom edge; can be omitted if narration suffices.

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Notes

• Assumed standard pedagogical pacing (≈8 s per major visual change) to fit the 60 s total.
• All labels use a large, sans‑serif font for readability on dark background.
• Projections are dashed lines to avoid clutter; they appear together with the point.
• The angle arc $\theta$ is shown as a thin white sector with the current value displayed near the arc.
• pauses at the five key angles are ~2 s each, giving enough time for students to read the displayed values.
• No on‑screen narration text is included; the specification assumes a separate audio track.
• The animation uses a single Manim Scene class; all elements are created within that scene.