Unit Circle Trainer

Master Quadrant I first, then conquer the rest through symmetry

1.Learn Q1

2.Memorize Q1

3.Symmetry

4.Mixed Drill

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.

0 / 15 correct in a row

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

sin + cos − tan −

Q1 — All

sin + cos + tan +

Q3 — Take

sin − cos − tan +

Q4 — Calculus

sin − cos + tan −

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.

0 / 20 correct in a row