Single large reference circle with key angles in degrees and radians: the standard trig reference for memorising sin, cos and tan of standard angles.

Free printable unit circle PDF. Large reference circle with 16 key angles in degrees and radians, the standard trig reference for memorising sin, cos and tan.

Unit Circle Paper | Free Printable PDF

Unit Circle Paper

Only the first quadrant (0° to 90°) has four 'new' angles: 0°, 30°, 45°, 60°, 90°. The other 12 angles are reflections and rotations of these. Memorise the sin and cos values for quadrant I, then derive the rest by symmetry and sign changes — far easier than rote memorisation of 16 separate values.

Memorise quadrant I, then mirror

The 30°, 45° and 60° coordinates come from two famous right triangles. The 30-60-90 triangle has sides 1, √3, 2, giving sin 30° = 1/2 and cos 30° = √3/2. The 45-45-90 has sides 1, 1, √2, giving sin 45° = cos 45° = √2/2. These two triangles unlock the whole unit circle.

Recognise 30-60-90 and 45-45-90 triangles

Calculus problems are almost always stated in radians (not degrees), because the derivative of sin x is cos x only when x is in radians. The radian labels on the inside of the circle make it easier to think in calc-friendly units.

Use the radian labels for calculus

Quadrant I (0–90°): both sin and cos positive. II (90–180°): sin positive, cos negative. III (180–270°): both negative. IV (270–360°): sin negative, cos positive. ASTC ('all students take calculus') is one of many mnemonics for remembering which functions are positive in each quadrant.