cos = horizontal, sin = vertical.
About the Unit Circle Explorer
The unit circle is where sine and cosine stop being calculator buttons and start being coordinates, but that connection is hard to see in a static textbook diagram. This tool puts a draggable angle on a live circle, drawing the reference triangle as the angle changes and reading out the exact coordinates, the radian measure, and all three trig ratios at once.
A snap-to-common-angle toggle locks the slider to the sixteen angles — multiples of 30° and 45° — that show up constantly in a trigonometry course, so it's easy to land exactly on 30°, 45°, or 120° instead of hunting for it by hand.
How to use it in your classroom
- Drag the angle slider to set a degree value from 0 to 360.
- Leave "Snap to common angles" checked to land precisely on the standard angles (multiples of 30° and 45°); uncheck it to explore any angle freely.
- Read the cosine, sine, and tangent values below the slider, along with the angle's radian measure.
- Watch the point and reference triangle move on the circle as the angle changes, and note how the point's coordinates always match (cos, sin).
Tips from the classroom
- Start with the snap toggle on and walk through the four quadrant angles (0°, 90°, 180°, 270°) before letting students explore freely — it anchors where each sign change happens.
- Point out that the point's coordinates are literally (cos, sin) at every angle, not a coincidence of this one example — that's the entire idea of the unit circle.
- Use 90° and 270° to show why tangent is undefined: cosine is zero there, and the tool displays that explicitly instead of an error or a huge number.
- Turn the snap toggle off for a quick estimation exercise — ask students to predict the sign of sine and cosine in each quadrant before checking the readout.
Frequently asked questions
Why does tangent sometimes say "undefined" instead of a number?
Tangent is sine divided by cosine, so whenever cosine is zero (at 90° and 270°), tangent has no defined value. The tool reports that directly rather than showing an error or an enormous number.
What does the snap-to-common-angles toggle actually do?
With it checked, the slider's value is rounded to the nearest of sixteen standard angles (every 30° and 45° around the circle) so you land exactly on values like 45° or 210° instead of an angle like 46.7°.
Are the angle values in degrees or radians?
The slider and main display use degrees, since that's typically introduced first, but the radian equivalent is shown alongside it at all times.