Unit Circle Radians Explorer
Run the Unit Circle Radians Explorer MicroSim Fullscreen
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About This MicroSim
Sine and cosine are usually introduced twice: once as ratios in a triangle, and once as wavy curves on a graph. The connection between those two pictures is a rotating point, and this sim shows both at the same time so the connection is impossible to miss.
On the left, a point travels around a circle of radius 1. Two dashed lines drop from it: one to the horizontal axis, one to the vertical.
- The horizontal projection is \(\cos\theta\).
- The vertical projection is \(\sin\theta\).
On the right, those two projections are plotted against the angle. The curves grow only as far as the point has traveled, so the leading dots on the right are always the same two numbers as the dashed lines on the left.
The angle is shown in radians first, degrees second. That ordering is deliberate — every formula in the rest of this course takes radians.
Landmarks
The four quarter-turn positions are marked on both views and light up in orange as the point passes them:
| Angle | Degrees | cos | sin |
|---|---|---|---|
| 0 | 0° | 1 | 0 |
| π/2 | 90° | 0 | 1 |
| π | 180° | -1 | 0 |
| 3π/2 | 270° | 0 | -1 |
How to Use
- Drag the Angle slider slowly from 0 and watch both views move together.
- Stop at π/2. Read the cos and sin values off the circle, then confirm the leading dots on the right agree.
- Note where cosine peaks and where sine peaks. They are a quarter turn apart — that is the whole sine/cosine phase relationship, visible geometrically.
- Press Play and adjust the Rotation speed. Watch the curves fill in.
- Find the angles where sine is zero. What is the point doing on the circle at those moments?
Lesson Plan
Grade Level
Undergraduate (college junior/senior) or advanced high school
Duration
10 minutes
Prerequisites
- Coordinates on a plane
- A circle of radius 1 centered at the origin
Learning Objective
Students will be able to interpret the relationship between a point rotating around the unit circle, its angle in radians, and the resulting sine and cosine waveforms, and explain where each curve comes from geometrically.
Activities
- Read both views (4 min): At five chosen angles, students record cos and sin from the circle and verify against the plot.
- Quarter turns (3 min): Students fill in the landmark table from the sim rather than from memory.
- Explain the offset (3 min): Students explain, using the circle, why cosine leads sine by a quarter turn.
Assessment
Ask: "At what angle in radians is the point highest on the circle, and what are cos and sin there?" (π/2; cos = 0, sin = 1.)
Related Resources
References
- Unit circle — the definition of sine and cosine by coordinates.
- Radian — why angles are measured in arc lengths in signal processing.