Online Tool Store
Unit Circle Explorer
Explore the unit circle: how an angle maps to sine and cosine coordinates, where the exact values come from, and how signs change by quadrant.
🔒 This tool runs entirely in your browser. Your files are never uploaded to a server.
Math & Science
Unit Circle Explorer
Frontend preview — no upload or external service.
Angle values
150° = 5π/6: coordinates (-√3/2, 1/2), so cos is negative and sin positive — reference angle 30°, second quadrant.
How the Unit Circle Explorer works
- Enter an angle in degrees or radians.
- Read the coordinates: the x value is the cosine and the y value is the sine, by definition on a unit circle.
- Use the reference angle to see why 150° shares its magnitudes with 30° and differs only in sign.
The method
A point on the unit circle at angle θ has coordinates (cos θ, sin θ), so the whole of trigonometry follows from where that point sits.
x = cos θ, y = sin θ, tan θ = y / x
At 150° the point is (-√3/2, 1/2): the same magnitudes as 30°, with x negative because the second quadrant lies left of the axis.
FAQ
Why memorise the unit circle?
You do not need to. Learn the first quadrant's exact values and the reference-angle rule, and every other angle follows from a sign change.
Where is tangent undefined?
Wherever cosine is zero — at 90° and 270° — because tangent is the ratio y over x, and the denominator vanishes there.
Why radians rather than degrees?
Because the radian measures arc length on a unit circle, which makes calculus of trigonometric functions come out cleanly. Degrees are a historical convention.
How we compare
| Feature | Online Tool Store | A graphing calculator | A stats package |
|---|---|---|---|
| Exact values shown | ✓ | Static | Yes |
| Reference angle explained | ✓ | ✓ | Sometimes |
| Interactive and free | ✓ | ✗ | ✗ |
| Works offline once loaded | ✓ | ✓ | ✗ |
Unit Circle Explorer teaches the reference-angle shortcut rather than the full table, since one quadrant plus sign rules covers the whole circle.