Unit circle & exact values

cos is x, sin is y: On a circle of radius 1, the point at angle θ (measured anticlockwise from the positive x-axis) has coordinates (cos θ, sin θ). So cos θ is the x-coordinate and sin θ is the y-coordinate, and tan θ = sin θ / cos θ.

Why it matters: The unit circle lets sin, cos, tan work for any angle (including obtuse and beyond), which right-angled triangles can't do.

The special angles: You must know the exact sin, cos and tan of 0°, 30°, 45°, 60°, 90° (and their radian forms) — they appear on the non-calculator Paper 1.

sin and cos mirror each other: Notice sin and cos swap across 45°: sin 30° = cos 60° = ½. So you really only need one column plus the symmetry.

Which ratios are positive where: Going round the circle, CAST tells you which ratio is positive in each quadrant: All (Q1), Sin (Q2), Tan (Q3), Cos (Q4). Outside its quadrant, a ratio is negative.

Use it for signs, the acute angle for the value: Find the value from the matching acute angle, then attach the sign from CAST.

sin(180° − θ) = sin θ: Angles around the circle share values: sin(180° − θ) = sin θ (same sine, supplementary angle); cos(180° − θ) = −cos θ; cos(−θ) = cos θ. These let you find a second angle with the same sine — central to the ambiguous case.

An angle on a straight line: If an angle sits on a straight line with θ (so it's 180° − θ), its sine is the same — handy in geometry questions.