The Sine Rule
SOH-CAH-TOA only works in right-angled triangles. The sine rule is your first tool for any triangle.
In any triangle we label each side with the lower-case letter of the angle opposite it: side a faces angle A, and so on. The sine rule says these three fractions are always equal:
See it stay equal
Drag any corner. The triangle changes shape, the sides and angles change — but the three ratios stay locked together. Even better, that shared value equals 2R, the diameter of the circle that passes through all three corners (the circumcircle, shown dashed):
Drag any corner ▲
That circumcircle link is the real reason the rule is true — every side is a chord of the same circle, and a chord's length is its diameter times the sine of the angle it subtends.
When do you use it?
The sine rule needs a matching side-and-angle pair to get started. Reach for it when you know:
✓ Two angles + any side
(AAS / ASA) — find the third angle from "angles add to 180°", then any other side.
✓ Two sides + an angle opposite one of them
(SSA) — find the second angle. But watch out…
sin can give two valid angles (because, for example, sin 40° = sin 140°). Sometimes both make a real triangle, sometimes only one. Always check: do the angles still add up to less than 180°?Worked example
A triangle has angle A = 40°, angle B = 75°, and the side between them isn't known — but side a = 8 cm (opposite A). Find side b.
b = a × sin Bsin A
b = 8 × sin 75°sin 40°
b = 8 × 0.9660.643 ≈ 12.0 cm
That's the whole method: write the rule, slot in the pair you know, solve for the one you want.