Use one bounded math branch where the complex plane, unit-circle rotation, polar coordinates, trig identities, inverse-angle reasoning, and motion traced from equations all stay tied to the same coordinate language.
This topic deepens the math subject without exploding the curriculum tree. Complex Numbers on the Plane keeps real part, imaginary part, magnitude, argument, addition, and multiplication readable on one plane, Unit Circle / Sine and Cosine from Rotation turns that same plane into live projections, Polar Coordinates / Radius and Angle keeps one radius-angle point tied to the same x and y geometry, Trig Identities from Unit-Circle Geometry shows how the core identities are forced by the same point on the same circle, Inverse Trig / Angle from Ratio keeps ratio recovery and quadrant checks on that same plane, and Parametric Curves / Motion from Equations finally reuses the coordinate pair while a moving point traces a path from x(t) and y(t).
Best first concepts
The topic page keeps these starts in their own compact row so the first screen is about orientation and next action, not stacked feature cards.
Read complex numbers as points and vectors on one plane, then keep addition and multiplication geometric instead of symbolic-only.
Complex plane geometry
Strong first stop for getting into this topic without scanning the whole library.
Keep one rotating point, its x and y projections, and the sine-cosine traces linked so the unit circle becomes the live source of both functions.
Rotation and projections
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Keep one point visible in polar and Cartesian views at the same time so radius and angle turn directly into x and y on the plane.
Radius-angle coordinates
Strong first stop for getting into this topic without scanning the whole library.
Keep one rotating point and its projections visible so the core trig identities stay tied to geometry instead of detached symbol rules.
Trig geometry on the plane
Strong first stop for getting into this topic without scanning the whole library.
Keep one polar point and its coordinate signs visible so inverse trig becomes angle-from-ratio reasoning with quadrant checks instead of a calculator-only output.
Angle recovery from x and y
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Keep x(t), y(t), the traced path, and the moving point visible together so shape and traversal stay distinct.
Plane motion from equations
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Specific learning goals
These goal cards stay authored and transparent. They reuse the current topic page, starter tracks, guided collections, concept bundles, and progress cues instead of adding a separate recommendation system on top of this branch.
Use the complex-and-parametric topic route, the new lesson set, the compact starter track, and the vectors topic page so the plane language widens from complex numbers into unit-circle and polar-coordinate geometry, then deepens into trig identities and inverse-angle reasoning before motion.
Primary move
Open topic route
No saved progress yet inside Complex Numbers and Parametric Motion.
Entry diagnostic
Start from the opening step
No saved diagnostic checks are available yet, so the opening step is still the best entry into the collection.
Reuses the guided collection entry for Complex and Parametric Motion Lesson Set, with 0 of 3 probes already ready.
No saved progress yet inside Complex Numbers and Parametric Motion.
Open the complex-and-parametric topic route is the next guided collection step.
Complex Numbers on the Plane opens this track and sets up the rest of the path.
No saved progress yet inside Vectors.
Grouped concept overview
Each group is authored in the topic catalog, but the actual concepts, progress badges, and track cues still come from the canonical concept metadata and shared progress model.
Group 01
Start with one plane where complex numbers behave as both points and vectors, and where multiplication can be read as scale plus turn.
把複數視為平面上的點與向量，再讓加法與乘法保持幾何直覺，而不只停留在符號操作。
Strong first stop for getting into this topic without scanning the whole library.
Group 02
Then keep one rotating point and one radius-angle point on the same plane so cosine, sine, and polar coordinates all become one linked geometry.
把旋轉點、x/y 投影與正弦餘弦曲線連在一起，讓單位圓成為這兩個函數的即時來源。
Strong first stop for getting into this topic without scanning the whole library.
讓同一個點同時出現在極座標與直角座標中，使半徑與角度如何變成 x、y 直接可見。
Strong first stop for getting into this topic without scanning the whole library.
Group 03
Once the point and its projections are stable, keep the same plane while the core identities and inverse-angle recovery come from the geometry instead of detached symbolic tricks.
讓旋轉點與其投影保持可見，使核心三角恆等式仍然連著幾何，而不是脫離圖像的規則。
Strong first stop for getting into this topic without scanning the whole library.
讓一個極座標點與其座標正負保持可見，使反三角函數變成帶象限檢查的由比值求角，而不只是計算器輸出。
Strong first stop for getting into this topic without scanning the whole library.
Group 04
Finally keep the same plane while x(t) and y(t) drive one moving point and make the difference between path and traversal visible.
把 x(t)、y(t)、描出的路徑與移動點同時顯示，清楚分開形狀本身與沿著曲線前進的方式。
Strong first stop for getting into this topic without scanning the whole library.