Q.Find the trigonometric functions of $0°, 30°, 45°, 60°, 150°, 180°, 210°, 300°, 330°, −30°, −45°, −60°, −90°, −120°, −225°, −240°, −270°, −315°$.

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 immediately gives $\sin(-\theta) = -\sin\theta$, $\cos(-\theta) = \cos\theta$, and from these $\tan(-\theta) = -\tan\theta$, $\cot(-\theta) = -\cot\theta$, $\sec(-\theta) = \sec\theta$, $\text{cosec}(-\theta) = -\text{cosec}\,\theta$. Together with the quadrant sign rule, these specific-angle and negative-angle results let every angle that is a simple multiple of 30°, 45°, or 60° — positive or negative — be evaluated exactly by hand.