Unit Circle Animation: Connect Sine and Cosine
Rotate a point around the unit circle, read its sine and cosine, and watch its height trace a sine wave. Pause at key angles and check the coordinates.
Trace the circle and its sine wave
Start on the right, at 0°. As the point rotates counterclockwise, its horizontal coordinate is cosine and its vertical coordinate is sine. The wave copies that vertical coordinate.
θ = 0° (0.000 radians). cos(θ) = 1.000; sin(θ) = 0.000.
Values are rounded to three decimal places. Nothing plays automatically; use the slider for a still-frame explanation.
Read sine and cosine as coordinates
A unit circle has radius 1 and center (0, 0). Measure θ counterclockwise from the positive horizontal axis. The point on the circle has coordinates (cos θ, sin θ): cosine tells you how far right or left the point is, and sine tells you how far above or below the center it is.
The animated graph uses θ on its horizontal axis and the point's vertical coordinate on its vertical axis. Its moving marker has the same vertical value as the circle's point. On small screens the plots stack vertically; the dotted guides still mark matching values. One complete turn produces one period of the sine curve. Cosine is shown by the point's horizontal coordinate and the numeric readout; this demonstration does not draw a second cosine curve.
Check four quarter-turns
| Angle | Radians | Point (cos θ, sin θ) | What to notice |
|---|---|---|---|
| 0° | 0 | (1, 0) | Start on the right; the sine curve starts at zero. |
| 90° | π/2 | (0, 1) | At the top, sine reaches its maximum. |
| 180° | π | (−1, 0) | On the left, sine is zero while cosine is negative. |
| 270° | 3π/2 | (0, −1) | At the bottom, sine reaches its minimum. |
| 360° | 2π | (1, 0) | Back at the starting point after one full period. |
Use the angle buttons to jump to the first four checkpoints. Move the slider to its end for 360°. These exact checkpoint values are more useful for checking the model than judging a nearly horizontal or vertical line by eye.
Explain signs before memorizing values
In the upper half of the circle, the point's vertical coordinate is positive, so sine is positive. In the lower half, sine is negative. Cosine is positive in the right half and negative in the left half.
Pause at 135°. The point is above and left of the center, so sine must be positive and cosine negative. At 225°, both coordinates are negative. The radius remains 1 in every quadrant; a negative coordinate describes direction, not a negative distance.
Why steady rotation does not make a triangle wave
The point moves around the circle at a steady angular rate during playback. Its height changes quickly near the horizontal axis and slowly near the top and bottom. The resulting height-versus-angle graph curves smoothly instead of rising and falling at a constant rate.
Ask a learner to compare a small turn near 0° with the same turn near 90°. The second turn changes the height much less. This connects the circle to the shape of a sine wave without requiring a derivative lesson.
Turn the demonstration into a teaching sequence
- Start with a still frame at 0° and establish what the two coordinates mean.
- Advance to 90° and ask the learner to predict the wave's height before showing the readout.
- Play a complete rotation. Follow the dotted guides to see the same vertical value represented in two places.
- Pause at a non-checkpoint angle and ask for the signs of both coordinates.
- Return to 360° and connect a full turn to the period 2π radians.
If you record a lesson, introduce the circle before the graph and avoid reading every changing decimal aloud. Use the checkpoint table as captions or still frames. The playback on this page is an interactive illustration; it does not export a video or generate a new scene from a prompt.
Build your next math animation
Use the math animation creation guide to turn a learning objective into a storyboard, check the mathematics, and plan the final explanation. The Pythagorean theorem animation illustrates an area proof instead of a changing coordinate. Return to animation examples to choose another pattern.
For the underlying definitions and exact values, see OpenStax: Unit Circle, Sine and Cosine Functions. For another interactive treatment, explore GeoGebra's introduction to trigonometry. The diagram on this page is an original implementation using those standard mathematical relationships.