HomeConcept libraryMathComplex Numbers and Parametric MotionComplex and Parametric Motion

Math Complex Numbers and Parametric MotionIntroStarter track

Concept module

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.

Interactive lab

Keep the stage, immediate control feedback, and linked study tools in one working bench.

Unit circle rotation

Keep one rotating point, its x and y projections, and the sine-cosine traces tied to the same angle so the unit circle reads as a live source for both functions.

Drag the point or use the phase slider

Controls

Adjust the live parameters and watch the bench respond.

Angular speed

ω

1 rad/s

Controls how quickly the point rotates around the unit circle.

Phase

ϕ

0.18 rad

Sets the starting angle before time begins to advance.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

More presets

Presets

Time

0.00 s / 6.28 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s6.28 s

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 2

When the point moves left or right, the cosine trace moves with the horizontal projection because they are the same x-value.

Graphs

Switch graph views without breaking the live stage and time link.

Cosine and sine traces

One graph shows cosine and sine changing together over time, and a second graph shows the same angle increasing over time.

time: 0 to 6.28projection value: -1.15 to 1.15

cos(θ)sin(θ)

Hover or scrub to link the graph back to the stage.time / projection value

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

ω

Angular speed

1 rad/s

Controls how quickly the point turns, so the angle graph steepens and the sine-cosine traces cycle faster.

Graph: Cosine and sine tracesGraph: Angle vs timeOverlay: Rotation trailOverlay: Angle marker

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Keep the circle and the traces visible together.

ObservationPrompt 1 of 2

Graph: Cosine and sine traces

When the point moves left or right, the cosine trace moves with the horizontal projection because they are the same x-value.

Control: PhaseGraph: Cosine and sine tracesOverlay: Projection guidesEquation

ϕ

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

4 visible

Overlay focus

Projection guides

Project the current point back to the x and y axes.

What to notice

The x-axis projection is exactly cosine and the y-axis projection is exactly sine.

Why it matters

It keeps the graph traces anchored to the same point on the circle.

Control: PhaseGraph: Cosine and sine tracesEquation

ϕ

Challenge mode

Use the live circle, the projection traces, and the sign map together. This checkpoint is about reading the sign change from the bench instead of memorizing a quadrant table in isolation.

0/1 solved

ConditionCore

4 of 5 checks

Quadrant II sign checkpoint

Start from the axis-crossing view, then push the point just into Quadrant II so cosine has flipped negative while sine is still strongly positive. Keep the projection guides and sign map on so the crossing stays visible.

Graph-linkedGuided start2 hints

Suggested start

Stay close to the top crossing so the horizontal projection has only just changed sign.

Matched

Open the Cosine and sine traces graph.

Cosine and sine traces

Matched

Keep the Projection guides visible.

Matched

Keep the Quadrant sign map visible.

Matched

Keep the Angle marker visible.

Pending

Set the starting angle between about

1.62

and

1.78 rad

so the point sits just into Quadrant II.

0.18 rad

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0 s, the unit-circle point sits in Quadrant I at theta = 10.31°. The horizontal projection is cos(theta) = 0.98, the vertical projection is sin(theta) = 0.18, and one full turn takes 6.28 s.

Equation detailsDeeper interpretation, notes, and worked variable context.

Point on the unit circle

The live point selected by angle theta has cosine as its x-coordinate and sine as its y-coordinate.

(x, y) = (cos θ, sin θ)

ϕ

Phase 0.18 rad

Angle in time

The angle grows steadily in time when the rotation rate stays constant.

θ (t) = ω t + ϕ

\omega controls how quickly the point sweeps through the circle.

\phi sets the starting angle.

ω

Angular speed 1 rad/s

ϕ

Phase 0.18 rad

Sign logic from the axes

The signs follow the side of each axis where the projection lands.

sgn (cos θ) = sgn (x), sgn (sin θ) = sgn (y)

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 1 compact task ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Jump to a named bench state or copy the one you are looking at now. Shared links reopen the same controls, graph, overlays, and compare context.

This bench opened from a setup link. Any new copy will reflect the state you see now.

Current bench

Reference turn preset

This bench still matches one named preset, so the copied link will reopen that same starting point along with the current graph, overlays, and inspect context.

Open default bench

Featured setups

Projection link

Open a clean counterclockwise turn with the guides and sign map visible.

Axis crossing

Start near the top of the circle and watch one projection drop to zero on the axis.

Saved setups

Premium keeps named exact-state study setups in your account while stable concept links stay public below.

Checking saved setup access.

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Copy current setup

Stable concept and section links stay public below while exact-state setup sharing stays behind premium.

Stable links

Starter track

Step 2 of 40 / 4 complete

Complex and Parametric Motion

Earlier steps still set up Unit Circle / Sine and Cosine from Rotation.

1. Complex Numbers on the Plane2. Unit Circle / Sine and Cosine from Rotation3. Polar Coordinates / Radius and Angle4. Parametric Curves / Motion from Equations

Previous step: Complex Numbers on the Plane.

Start from track beginning

Short explanation

What the system is doing

The unit circle is easiest to trust when one rotating point, its horizontal and vertical projections, and the sine-cosine traces all stay visible together. This bench keeps those pieces tied to the same live angle instead of treating sine and cosine as detached graph rules.

Cosine is the x-coordinate because the horizontal projection is literally the point's shadow on the x-axis. Sine is the y-coordinate for the same reason on the y-axis. As the point moves through the quadrants, the signs change because the projections change side with the point.

Key ideas

01On the unit circle, the point selected by angle theta is the ordered pair (cos theta, sin theta).

02Cosine is the horizontal projection and sine is the vertical projection of the same rotating point.

03Quadrant sign changes are geometry first: left side means negative cosine, below the axis means negative sine.

Worked example

Read the full frozen walkthrough.

Frozen walkthrough

Use the current rotation state instead of a detached table of special angles.

Live worked examples are available on Premium. You can still read the full frozen walkthrough on the free tier.

View plans

Frozen valuesFrozen at t = 0.00 s

For the current time and rotation state, what point on the unit circle is the motion selecting?

t

Time

0 s

ω

Angular speed

1 rad/s

ϕ

Phase

0.18 rad

1. Read the current angle

The live angle comes from

θ = ω t + ϕ

, so the current rotation is at

θ = 0.18 rad

or about 10.31 deg.

2. Read the two projections from the same point

The horizontal projection gives

cos θ = 0.98

and the vertical projection gives

sin θ = 0.18

3. Write the current unit-circle point

So the rotating point is

(cos θ, sin θ) = (0.98, 0.18)

Current unit-circle point

(cos θ, sin θ) = (0.98, 0.18)

The positive cosine value matches the point staying on the right side of the unit circle.

Common misconception

Sine and cosine are separate graph tricks, so the unit circle picture is optional.

The graphs come from the same rotating point on the unit circle.

When the point moves left or right, cosine changes sign; when it moves above or below the x-axis, sine changes sign.

Mini challenge

If the point keeps rotating counterclockwise from Quadrant I into Quadrant II, which projection has to change sign first?

Prediction prompt

Decide whether the point crosses an axis that changes x first or an axis that changes y first.

Check your reasoning

Cosine changes sign first because the point crosses the y-axis before it reaches the x-axis again.

Crossing the y-axis flips the horizontal projection from positive to negative while the vertical projection stays above the x-axis.

Quick test

Misconception check

Question 1 of 2

Answer from the live circle and traces.

Which statement is correct on the unit circle?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows a unit circle with one rotating point, axis projections for cosine and sine, an angle marker, and a sign map for the four quadrants.

Graph summary

One graph shows cosine and sine changing together over time, and a second graph shows the same angle increasing over time.

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Keep the same angle language while radius turns into x and y on the planeComplex Numbers and Parametric Motion

Unit Circle / Sine and Cosine from Rotation

See each variable before you move it.

Projection guides

Quadrant II sign checkpoint

Complex and Parametric Motion

What the system is doing

Read the full frozen walkthrough.

For the current time and rotation state, what point on the unit circle is the motion selecting?

Which statement is correct on the unit circle?

Keep moving through the concept map.

Polar Coordinates / Radius and Angle

Parametric Curves / Motion from Equations

Uniform Circular Motion