This simulator sweeps a radius around a circle and tracks its horizontal projection x = r cos θ and vertical projection y = r sin θ as continuously linked quantities. Change the angle, angular velocity or radius and watch how a rotation turns into the sine and cosine waveforms.
• A real-time 3D-staged view of the rotating radius, its perpendicular horizontal (gold) and vertical (magenta) projection segments, linked sine and cosine waveform ribbons that share the angular coordinate, and coordinate/angular reference axes, with home view, focus-selected-part, auto-rotate, expand and toggleable labels. • Three live sliders: angle θ (0° to 720°, two full turns), angular velocity ω (−120° to 120° per second, sign sets rotation direction), and radius r (0.5 to 2 coordinate units). • Play/pause, single-step and larger-step time controls, plus a playback-speed selector from 0.1x to 60x laboratory speed. • Two actions: start rotation and stop rotation. • Six live metrics: horizontal projection x, vertical projection y, sin θ, cos θ, the angle in radians, and tangential speed. • A Curves & measurements tab with a sine-and-cosine chart and an angle/coordinate-reference chart, the full model equations, and snapshot measurements. • An Experiments tab with four guided scenarios (quarter turn, third quadrant, double the radius, reverse rotation), a model-verification bench of independent automated checks, and a timestamped event log with a copyable trial report. • A Learn & assess tab with four guided lessons, a two-question knowledge-check quiz, and a written scope/reference statement linking to OpenStax Precalculus.
At any instant, the rotating radius's endpoint sits at (r cos θ, r sin θ). On a unit circle those coordinates are exactly cosine and sine; at other radii, both coordinates scale by r while the trigonometric ratios themselves stay unchanged, since sine and cosine are defined as ratios (y/r and x/r). Sine is positive above the horizontal axis and cosine is positive to the right of the vertical axis — signs are set by quadrant position, not by which direction the radius is rotating.
Plotting each projection against the swept angle is what produces a sinusoidal wave: a constant angular velocity produces a smooth, evenly-paced sine or cosine curve, while a negative angular velocity runs the same waveform backward in time rather than producing a different shape. Converting between the two units matters for speed: tangential speed is v = r|ω| only once ω is expressed in radians per second, since a radian directly measures arc length divided by radius.
The Curves & measurements tab overlays the sine and cosine traces against angle, letting you see their quarter-cycle phase offset and confirm that each ranges between −r and +r. The equations panel states x = r cos θ, y = r sin θ, the degree-to-radian conversion, the identity x² + y² = r², tangential speed v = r|ωrad|, and the period T = 360°/|ωdeg| for nonzero constant angular velocity.
The four built-in experiments walk through reading both projections at a quarter turn (90°), checking the matched-magnitude negative coordinates in the third quadrant (225°), confirming that doubling the radius scales the coordinates but not the sine or cosine ratios themselves, and reversing the rotation direction to see the endpoint trace the same functions backward. The model-verification bench in the Experiments tab independently checks quadrant coordinates, radius scaling and other exact circle-kinematics relationships against fresh model instances, leaving your current trial untouched.
At angle θ, the rotating radius's endpoint has horizontal coordinate x = r cos θ and vertical coordinate y = r sin θ, where r is the current radius. On a unit circle (r = 1) these coordinates equal cosine and sine directly; at any other radius they are that radius times the corresponding ratio.
No. Sine and cosine are ratios — y/r and x/r — so they depend only on the angle, not the radius. Doubling the radius doubles the actual coordinates x and y, but the sine and cosine values themselves stay exactly the same, which is one of the four guided experiments in this simulator.
Angular velocity ω sets both the speed and direction of rotation: a negative ω sweeps the angle backward in time through the same sequence of quadrant positions. Because sine and cosine are still evaluated at whatever angle the radius currently occupies, the projections trace the identical sinusoidal functions — just played in reverse — rather than becoming a different family of curves.
This is an exact circle-kinematics and trigonometry teaching model. The 3D presentation places genuinely two-dimensional plots on labeled display planes for viewing convenience; it does not represent an additional physical dimension, and the swept angle simply wraps back to zero after two full turns (720°) rather than continuing to accumulate indefinitely.