What quantity did you change?
Labs / Mathematics
Unit Circle Unwrapped
A point rides around a circle of radius one, and its height gets projected out to the right to draw a wave. Drag the point, switch between sine, cosine & tangent, and watch how spinning in a circle and rolling out a wave are secretly the same motion.
Labs / Mathematics / Unit Circle Unwrapped
Unit Circle Unwrapped
A point rides around a circle of radius one, and its height gets projected out to the right to draw a wave. Drag the point, switch between sine, cosine & tangent, and watch how spinning in a circle and rolling out a wave are secretly the same motion.
Why it matters
Why explore this?
Mathematics helps us describe patterns precisely and predict what comes next. This model turns an abstract relationship into something you can change and observe.
Explore
Try the model
θ 30.0°(cos, sin) (0.866, 0.500)sin 0.500cos 0.866tan 0.577
Investigate
Questions to test
Predict first, try it in the model, then use what you observe to answer.
What to try
- Where on the circle is the point when the sine wave hits its very top? What angle is that?
- Switch to cosine. Why does its wave start at 1 instead of 0 — what is the point's x-coordinate at 0°?
- Where does the tangent wave shoot off toward infinity, and what is cosine doing at exactly those angles?
- Flip to radians. How far around the circle is one full wavelength — and why is it 2π?
- Drag the point past 180°. Which of sin, cos & tan go negative, and can you predict it from the (x, y) coordinates?
Explain
Make sense of what happened
State the pattern in words, test it with another example, and then connect it to the mathematical rule or representation shown in the model.
Check
Before you leave
What pattern did you observe?
Does the pattern work for another example?
If you can answer all three using evidence from the model, you understand the main idea. If not, return to Explore and change one setting at a time.