Many trigonometric values used throughout mathematics come from a small set of specific angles — 0°, 30°, 45°, 60°, 90°, and angles built from them such as 120°, 225°, and negative angles like −60° — whose exact values can be read off geometrically rather than approximated. The method is always the same: place the angle in standard position, drop a perpendicular from the point P on the unit circle to the x-axis, and use a recognisable 30°-60°-90° or 45°-45°-90° triangle to find the coordinates of P exactly (e.g. for 120°, OQ = 1/2 and PQ = √3/2, giving P(−1/2, √3/2) since 120° is in quadrant II). At the axis angles 0°, 90°, 180°, 270°, 360° the point P sits exactly on an axis, e.g. P(1,0) at 0° and P(0,1) at 90°, which is why some functions (like tan90°, sec90°) are undefined there — the coordinate needed in the denominator is zero. Negative angles are handled by reflecting P across the x-axis: the point for −θ is the mirror image of the point for θ, which immediatel …