Starter track
Follow the authored sequence, or switch to recap mode for a faster review of the same path.
Start with complex numbers as points on one plane, turn that plane into unit-circle and polar-coordinate geometry, deepen that same bench into trig identities and inverse-angle reasoning, then carry the coordinate language into motion traced from x(t) and y(t).
Entry diagnostic
Reuse the complex-plane quick test plus the unit-circle sign and polar x-y checkpoints to decide whether to start from plane geometry or jump straight to the trig-geometry deepening before motion.
Check the plane-to-rotation bridge first
No saved diagnostic checks are available yet, so the opening concept is still the best place to start.
Uses the same local-first quick tests, checkpoint challenges, and track history already saved in this browser.
Check whether points, vectors, magnitude, and argument already read as one object on the same plane.
No saved quick-test result yet.
Start from the axis-crossing view, then move just into Quadrant II so cosine is negative while sine is still clearly positive. Keep the guides and sign map on so you can see exactly why the signs differ.
No saved checkpoint attempt yet.
Build a point in Quadrant II where the leftward x-projection has larger magnitude than the upward y-projection. Keep the coordinate guides on so r, θ, x, and y stay tied to the same point.
No saved checkpoint attempt yet.
About this track
Keep the first scan focused on the next lesson. Open the authored rationale and shared-framework notes only when you need them.
Why this order
Complex Numbers on the Plane comes first because it turns the plane itself into a richer mathematical object before any motion story is layered onto it. Unit Circle / Sine and Cosine from Rotation comes next because it keeps the same plane while turning x and y into live projections from one angle. Polar Coordinates / Radius and Angle then keeps that same angle language while making radius and Cartesian components readable together. Trig Identities from Unit-Circle Geometry uses that same point to justify the core identity without leaving the bench, Inverse Trig / Angle from Ratio keeps ratio recovery and quadrant checks on the same plane, and Parametric Curves / Motion from Equations finally reuses the same coordinate pair while separating the traced path from the motion along it.
Shared concept pages
Compare mode, prediction mode, quick test, worked examples, guided overlays, challenge mode, and read-next cues stay on the concept pages. The track only decides the guided order and the next recommended stop.
Guided path
Checkpoint cards reuse the authored challenge entries already living on the concept pages.
Read complex numbers as points and vectors on one plane, then keep addition and multiplication geometric instead of symbolic-only.
Start here before moving into Unit Circle / Sine and Cosine from Rotation.
In multiplication view, choose w so z · w lands almost on the positive imaginary axis while the multiplier magnitude stays close to 1.
Finish Complex Numbers on the Plane first. This checkpoint ties together Complex numbers through Turn the product onto the positive imaginary axis.
Pause here after Complex Numbers on the Plane before moving into Unit Circle / Sine and Cosine from Rotation.
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.
Builds on Complex Numbers on the Plane before setting up Polar Coordinates / Radius and Angle.
Start from the axis-crossing view, then move just into Quadrant II so cosine is negative while sine is still clearly positive. Keep the guides and sign map on so you can see exactly why the signs differ.
Finish Unit Circle / Sine and Cosine from Rotation first. This checkpoint ties together Complex numbers and Unit circle rotation through Catch the first sign flip.
Pause here after Unit Circle / Sine and Cosine from Rotation before moving into Polar Coordinates / Radius and Angle.
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.
Builds on Unit Circle / Sine and Cosine from Rotation before setting up Trig Identities from Unit-Circle Geometry.
Keep one rotating point and its projections visible so the core trig identities stay tied to geometry instead of detached symbol rules.
Builds on Polar Coordinates / Radius and Angle before setting up Inverse Trig / Angle from Ratio.
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.
Builds on Trig Identities from Unit-Circle Geometry before setting up Parametric Curves / Motion from Equations.
Build a point with negative $y / x$ whose full angle is clearly in Quadrant II, not Quadrant IV. Keep the angle-recovery graph open so the principal-angle and actual-angle curves visibly disagree.
Finish Inverse Trig / Angle from Ratio first. This checkpoint ties together Complex numbers, Unit circle rotation, Polar coordinates, Trig identities, and Inverse trig through Quadrant II recovery checkpoint.
Pause here after Inverse Trig / Angle from Ratio before moving into Parametric Curves / Motion from Equations.
Keep x(t), y(t), the traced path, and the moving point visible together so shape and traversal stay distinct.
Capstone step after Inverse Trig / Angle from Ratio.
Make the curve clearly taller than it is wide, then catch the point near the y-axis during a fast part of the motion.
Finish Parametric Curves / Motion from Equations first. This checkpoint ties together Complex numbers, Unit circle rotation, Polar coordinates, Trig identities, Inverse trig, and Parametric curves through Catch a fast point on a tall curve.
Final checkpoint that closes the authored track after Parametric Curves / Motion from Equations.
