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Math · Complex Numbers and Parametric Motion

Unit Circle / Sine and Cosine from Rotation

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Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Polar Coordinates / Radius and AngleKeep angle as direction while radius changes the size of the projection story.

Key takeaway

On the unit circle, the angle θ selects the point (cos θ, sin θ).
Cosine is the horizontal projection and sine is the vertical projection of that same point.
Quadrant signs come from axis crossings: left of the y-axis means negative cosine, and below the x-axis means negative sine.
Changing angular speed changes how fast the traces are drawn, not the coordinate rule at each angle.

Common misconception

Do not treat sine, cosine, and quadrant signs as separate memorized tables. They are all readings of one rotating point and its projections.

The traces come from the same rotating point on the circle.

Keep this idea moving

Open the next concept, route, or track only when you want the current model to widen into a larger branch.

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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.

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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.

See the same projection story inside a physics rotation benchMechanics

Uniform Circular Motion

Track a particle moving at constant speed around a circle and connect radius, angular speed, tangential speed, centripetal acceleration, and the inward-force requirement to the same live state.

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More practice optionsReview on the bench · Worked examples

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Practice optionWorked examplesCompare against solved walkthroughs after you try the setup yourself.
Practice optionFree play on the benchExperiment freely with the live controls before moving to another concept.

Reference and support

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Return here when you want the slower reference pass.

Open reference and support

Equation snapshotPoint on the unit circle · Angle in timeShow 3 equations

Start with the point (cos θ, sin θ), then use θ(t) only to explain how the same point moves and why the projection signs change at the axes.

Point on the unit circle
$(x, y) = (cos θ, sin θ)$
Because the radius is 1, the coordinates of the point selected by angle theta are exactly cosine theta and sine theta.
Angle in time
$θ (t) = ω t + ϕ$
With constant angular speed, the angle changes linearly in time.
Sign logic from the axes
$sgn (cos θ) = sgn (x), sgn (sin θ) = sgn (y)$
The sign of $cos θ$ matches the sign of x, and the sign of $sin θ$ matches the sign of y.

Why it behaves this way

Explanation

The unit circle is easier to understand when one rotating point, its projections onto the axes, and the sine-cosine traces all stay visible together. This bench ties every view to the same live angle, so the graphs read as a record of the motion rather than as separate rules to memorize.

Because the circle has radius 1, the x-coordinate of the point is exactly $cos θ$ and the y-coordinate is exactly $sin θ$ . When the point moves left of the y-axis, cosine becomes negative; when it moves below the x-axis, sine becomes negative.

Key ideas

01On the unit circle, angle

θ

selects the point

(cos θ, sin θ)

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

03Sign changes come from position on the axes: left gives negative cosine, and below the x-axis gives negative sine.

Worked examples

Open examples when you want to see the same idea walked through step by step.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the current rotation state and read the geometry first. The same live angle sets the point on the circle, the axis projections, and the graph markers.

Supporter unlocks saved study tools, exact-state sharing, and the richer review surfaces that support this guided flow.

View plans

Example 1 of 2

Frozen valuesFrozen at 0.00

At the current time, which point on the unit circle is selected, and what does that tell you about $cos θ$ and $sin θ$ ?

t

Time

0 s

ω

Angular speed

1 rad/s

ϕ

Phase

0.18 rad

1. Find 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 cosine and sine from the projections

The horizontal projection gives

cos θ = 0.98

and the vertical projection gives

sin θ = 0.18

3. Write the point as an ordered pair

So the rotating point is

(cos θ, sin θ) = (0.98, 0.18)

Current point on the unit circle

(cos θ, sin θ) = (0.98, 0.18)

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

Quick test

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Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows a unit circle with one rotating point, projection guides to the x- and y-axes, an angle marker, a swept arc, and a sign map for the four quadrants.

Graph summary

One graph shows cosine and sine changing over time, and the other shows the angle changing over time. Both graphs are linked to the same moving point on the circle.

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Featured setups

See projections live

Start with the guides and sign map on so one angle, two projections, and two traces stay connected.

Watch an axis crossing

Start near the top of the circle so one projection is close to zero and a sign flip is easy to catch.

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Progress

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Starter track

Step 2 of 6

Complex and Parametric Motion

Unit Circle / Sine and Cosine from Rotation appears later in this track, so it is cleaner to start from the beginning first.

Previous step: Complex Numbers on the Plane

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