the law of sines
Right-triangle trigonometry runs out the moment a triangle has no right angle. The law of sines is one of the two tools that rescue these oblique triangles, and it is the one to reach for whenever your given information pairs an angle with the side facing it.
In any triangle, label the angles A, B, C and the sides opposite them a, b, c. The law of sines says these three ratios are all equal: a / sin A = b / sin B = c / sin C. (More fully, this common value equals the diameter 2R of the triangle's circumscribed circle.) In words, the bigger the angle, the longer the side across from it, in exactly this proportional way. To use it you need a complete angle-side pair plus one more measurement; the law then solves for the matching unknown. It fits two friendly situations: two angles and any side (called ASA or AAS), and two sides with an angle opposite one of them (called SSA).
The law of sines is the natural partner to the law of cosines: use sines when an angle and its opposite side travel together, and cosines when they do not. One caution dominates: the SSA case can have two valid triangles, one, or none — the so-called ambiguous case — because the equation sin X = (some value) has two angles between 0 and 180 degrees. Always check whether a second, obtuse solution also fits.
In a triangle, angle A = 40 degrees, angle B = 75 degrees, and side a = 10 (opposite A). Then side b = a x sin B / sin A = 10 x sin 75 degrees / sin 40 degrees ≈ 10 x 0.966 / 0.643 ≈ 15.0. Angle C = 180 - 40 - 75 = 65 degrees, and c follows the same way.
Each side divided by the sine of its opposite angle gives the same number — that is the whole rule.
The law of sines needs a matched angle-side pair to start; if you only know all three sides, or two sides and the included angle, it cannot get going and you must use the law of cosines instead.