Exact values of trigonometric functions of widely used angles

For a handful of frequently used angles, the trigonometric ratios work out to clean, exact values rather than messy decimals — worth memorising rather than recomputing each time.

A few things worth noticing:

• Every entry is an exact value (a whole number, a simple fraction, or a fraction with a square root) — never a rounded decimal.
• $\sin30^\circ=\cos60^\circ=\dfrac12$ and $\sin60^\circ=\cos30^\circ=\dfrac{\sqrt3}{2}$ — not a coincidence, but an early hint of the general complementary-angle relationship $\sin\theta=\cos(90^\circ-\theta)$.
• The reciprocal ratios $\csc\theta,\sec\theta,\cot\theta$ at each of these angles can always be read off the same table, since they are just $1/\sin\theta$, $1/\cos\theta$, $1/\tan\theta$. …