Unit Circle Trainer
Master Quadrant I first, then conquer the rest through symmetry
Stage 1 — Learn Quadrant I
Every other angle on the unit circle is just a reflection of one of these five. Study this table until you can recite it cold — angle, radians, sine, cosine, and tangent. You cannot continue until you study and acknowledge the table below.
| Degrees | Radians | cos θ | sin θ | tan θ |
|---|
Memory tip: cosine values count down √4/2, √3/2, √2/2, √1/2, √0/2 from 0° to 90°. Sine values count up the same way. Tangent = sine ÷ cosine.
Stage 2 — Memorize Quadrant I
You must answer 15 questions in a row from Quadrant I. One wrong answer resets the streak. When you reach a perfect round, the next stage unlocks.
Stage 3 — Symmetry of the Other Quadrants
Every angle outside Quadrant I has a reference angle in Q1 — the acute angle it makes with the x-axis. The magnitude of sin, cos, and tan is identical to that reference angle. Only the signs change, depending on the quadrant.
Reference angle by quadrant
- Q1: ref = θ
- Q2: ref = 180° − θ (or π − θ)
- Q3: ref = θ − 180° (or θ − π)
- Q4: ref = 360° − θ (or 2π − θ)
Signs by quadrant — All Students Take Calculus
Q2 — Students
Q1 — All
Q3 — Take
Q4 — Calculus
Recipe for any angle: (1) identify quadrant, (2) find reference angle in Q1, (3) look up the Q1 magnitude, (4) apply the quadrant’s sign.
Stage 4 — Mixed Quadrant Drill
Now you’ll be asked about angles in all four quadrants. Use your Q1 values plus the sign rules. You need 20 correct in a row. Some questions will explicitly ask for the reference angle to keep the symmetry sharp.