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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

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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.

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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

Why it behaves this way

Explanation

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.

Frozen walkthrough

Step through the frozen example

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

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Frozen valuesFrozen at 0.00

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?

Make a prediction before you reveal the next step.

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

Check your reasoning against the live bench.

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?

Use the live bench to test the result before moving on.

Accessibility

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.