§1The six modes
- deg ↔ rad: convert a special angle between degrees and radians, such as 150° to 5π/6 or 7π/4 to 315°.
- coordinates: give the point (cos θ, sin θ) for an angle, for example (−√3/2, −1/2) at 210°.
- sin · cos · tan: give one exact value, including “undefined” for tan 90°.
- quadrant & sign: say where an angle ends and whether a function is positive, negative, zero or undefined there.
- fill the circle: drag radian, degree or coordinate labels onto a blank unit circle. It is the classic blank circle quiz, checked as you go.
- speed run: 60 seconds of mixed multiple-choice questions. Your best score for each level is saved.
§2The three levels
Quadrant I uses only 0°, 30°, 45°, 60° and 90°. These five angles carry every value on the circle, so they come first. Full circle uses all 16 special angles from 0° to 330°. Beyond 360° uses coterminal angles such as −210° or 495°, which is where most test questions try to trip students up: you have to find the matching angle in one turn before you can answer.
§3Typing exact answers
Answers are checked by value, not by spelling, so equivalent forms count. For tan 30°, both √3/3 and 1/√3 are accepted. For a radian conversion, 5π/6, 5pi/6 and (5π)/6 all work. Use the keys under the box for √ and π, or type sqrt(3) and pi. Coordinates go in as a pair: (−√2/2, √2/2).
Decimals are not accepted as exact answers. If you type 0.866 for sin 60°, the practice tells you it is close and shows the exact value √3/2, because that is what a test will expect.
§4A practice plan that works
- Spend a few minutes on the interactive unit circle, moving between 30°, 150°, 210° and 330° to see that the same triangle repeats in every quadrant.
- Do Quadrant I in all modes until you miss nothing.
- Switch to the full circle, starting with quadrant & sign, then coordinates and values.
- Fill in the circle with radians, then with coordinates, on the full circle level.
- Finish each session with one speed run and try to beat your best.
Short daily sessions work better than one long one. Five minutes a day for a week is usually enough to know the circle by heart.
§5Getting ready for a unit circle test
Most unit circle tests ask for exact values without a calculator, conversions between degrees and radians, and a blank circle to fill in. Practice all three here, then print a blank unit circle and fill it in by hand, timing yourself. Writing it out once on paper the day before a test is a good final check. For angles outside one turn, the coterminal angle calculator and the reference angle calculator show the steps.
§6Common mistakes
- Swapping sine and cosine in coordinates. The point is (cos θ, sin θ): x first.
- Mixing up √3/2 and 1/2 between 30° and 60°. At 30° the point is wide and low, so cos is the big one (√3/2).
- Dropping signs in Quadrants II, III and IV. Decide the sign from the quadrant before you write the number.
- Treating −30° as 30°. It is coterminal with 330°, in Quadrant IV.
§7FAQ
Is my progress saved?
Yes, in your browser only (localStorage). There are no accounts, and nothing is sent to a server. Clearing your browser data or using a private window resets it.
Can I use it without a mouse?
Yes. Type answers and press Enter, use keys 1–5 for multiple choice, and in fill the circle select a label and press Enter on a point.
Why can’t I see the answer before I respond?
The correct answer is only worked out when you check, so it is not hidden anywhere in the page. Use “Show answer” if you are stuck.
Does it work on a phone?
Yes. All modes work with touch, and labels can be dragged onto the circle.