Skip to content
Open Model LabInteractive concept
HomeConcept libraryMathComplex Numbers and Parametric MotionComplex and Parametric Motion
MathComplex 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

Loading the live simulation bench.

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.

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.

This concept can keep using stable links while the saved-setups capability resolves for this browser.

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 60 / 6 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. Trig Identities from Unit-Circle Geometry+2 more steps

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?

Time

0 s

Angular speed

1 rad/s

Phase

0.18 rad

1. Read the current angle

The live angle comes from , so the current rotation is at or about 10.31 deg.

2. Read the two projections from the same point

The horizontal projection gives and the vertical projection gives .

3. Write the current unit-circle point

So the rotating point is .

Current unit-circle point

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.