Trigonometric Functions on the Unit Circle

Overview

A dark‑mode animation that builds the unit circle, sweeps an angle $\alpha$ from the positive $x$-axis, and visually defines $\cos\alpha$, $\sin\alpha$, $\tan\alpha$ and $\cot\alpha$ using perpendicular projections and tangent lines. The key takeaway is the geometric meaning of each trig function and the relationships $\tan\alpha=\frac{\sin\alpha}{\cos\alpha}$ and $\cot\alpha=\frac{\cos\alpha}{\sin\alpha}$.

---

Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Fade in a dark background, draw the white orthogonal axes $Oxy$ with arrowheads and label the origin $O$.
2	Unit Circle & Fixed Ray	~4s	Draw the unit circle centered at $O$ (radius 1). Highlight the fixed ray $OA$ on the positive $x$-axis ending at $A(1,0)$.
3	Sweeping Angle $\alpha$	~5s	Animate a rotating ray $OM$ from $OA$ to a variable position $M$ on the circle, drawing the angle arc and labeling it $\alpha$.
4	Cosine & Sine Projections	~6s	Drop a vertical line from $M$ to the $x$-axis meeting at $H$; highlight segment $OH$ in green as $\cos\alpha$. Drop a horizontal line from $M$ to the $y$-axis meeting at $K$; highlight segment $OK$ in orange as $\sin\alpha$.
5	Tangent $\tan\alpha$	~5s	Draw the tangent line at $A$ (vertical line $x=1$). Extend ray $OM$ until it meets this tangent at $T$; highlight $AT$ in purple and label its length as $\tan\alpha$.
6	Cotangent $\cot\alpha$	~5s	Draw the tangent line at $B(0,1)$ (horizontal line $y=1$). Extend ray $OM$ until it meets this tangent at $S$; highlight $BS$ in pink and label its length as $\cot\alpha$.
7	Summary Fade‑out	~2s	Fade out all auxiliary lines, leaving the circle, axes, and the four highlighted segments with brief labels (optional).

---

Layout

┌─────────────────────────────────────────────┐
│                                             │
│               MAIN (unit circle)            │
│                                             │
├─────────────────────────────────────────────┤
│  Caption (optional small text for phase)    │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Dark background, white orthogonal axes, unit circle, rotating ray $OM$, projection points $H,K$, tangent intersections $T,S$, highlighted segments (green, orange, purple, pink).	Primary visual focus throughout all phases.
Caption	Short phrase indicating the current concept, e.g., "Cosine = horizontal projection".	Shown only during phases 4‑6; can be omitted if space is needed.

---

Notes

• Assumptions: No explicit textual labels for points are required beyond the highlighted segments; the caption area provides brief context. The animation runs ~30 seconds total, well within the 30‑second guideline.
• All colors are chosen for contrast on a dark background: green for cosine, orange for sine, purple for tangent, pink for cotangent.
• The angle arc $\alpha$ is drawn with a thin white arc and a small label "$\alpha$" near its middle.
• The tangent at $A$ is the vertical line $x=1$; the tangent at $B$ is the horizontal line $y=1$.
• The scene will be implemented as a single Manim Scene class.
• No persistent on‑screen text is used except the optional caption; all mathematical meaning is conveyed visually.