The Sine Rule

You already know SOH-CAH-TOA — sine, cosine and tangent let you chase missing sides and angles around a triangle. But there's a catch nobody tells you at first: it only works in a right-angled triangle. The moment the triangle has no 90^\circ corner, SOH-CAH-TOA is stuck.

Real triangles are rarely so polite. The one a surveyor sights across a valley, the one a sailor draws between two lighthouses, the one your phone builds between GPS satellites — almost none of them have a right angle. The sine rule breaks the limit. It works in any triangle, and it says something beautifully simple: each side is tied to the sine of the angle opposite it, and the ratio is always the same.

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

The trick is the labelling: side a is opposite angle A, side b opposite B, side c opposite C. Big side, big angle across from it — divide one by the sine of the other and, in every triangle on Earth, you get the same number.

In any triangle, with each side opposite the angle of the same letter:

the ratios are all equal: \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C};
to find a side, use the side ÷ sin(opposite angle) form, e.g. b = \dfrac{a\,\sin B}{\sin A};
to find an angle, flip it upside down to \dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c};
and remember the angles of the triangle still sum to 180^\circ, so a missing angle often comes for free.

Labelling the triangle

Here is a scalene (no right angle, all sides different) triangle. Step through it to see how the sides a, b, c sit opposite the angles A, B, C.

Worked example 1 — find a missing side

A triangle has angle A = 40^\circ with its opposite side a = 8\text{ cm}, and angle B = 75^\circ. Find side b.

You have a complete pair — a with its opposite A — plus one more angle. That's exactly what the sine rule wants. Keep the unknown on top so no dividing-by-an-unknown is needed:

\frac{b}{\sin B} = \frac{a}{\sin A} \;\Longrightarrow\; b = \frac{a\,\sin B}{\sin A}

b = \frac{8 \times \sin 75^\circ}{\sin 40^\circ} = \frac{8 \times 0.9659}{0.6428} \approx 12.0\text{ cm}

Sanity check: B is bigger than A, so its opposite side ought to be longer than a = 8 — and 12.0 > 8. Good.

Worked example 2 — find a missing angle

Now you're given two sides and an angle that isn't between them: a = 12 opposite A = 65^\circ, and b = 9. Find angle B.

To find an angle, flip the rule so the sines sit on top:

\frac{\sin B}{b} = \frac{\sin A}{a} \;\Longrightarrow\; \sin B = \frac{b\,\sin A}{a}

\sin B = \frac{9 \times \sin 65^\circ}{12} = \frac{9 \times 0.9063}{12} \approx 0.6797

B = \sin^{-1}(0.6797) \approx 42.8^\circ

Again the check holds: b = 9 is shorter than a = 12, so its opposite angle should be smaller than 65^\circ — and 42.8^\circ is. (But see the "Watch out!" box below — inverse sine can hide a second answer.)

Worked example 3 — surveying across a river

A surveyor wants the distance to a tree on the far bank without swimming across. She walks a 50\text{ m} baseline along her own side and, with a theodolite, measures the angle to the tree from each end of the line: 62^\circ at one end and 74^\circ at the other. How far is the tree from the first station?

The two angles sit at the ends of the baseline, so the third angle (at the tree) is

180^\circ - 62^\circ - 74^\circ = 44^\circ.

The baseline (50\text{ m}) is opposite the tree's 44^\circ — that's our complete pair. The distance we want is opposite the 74^\circ angle, so

d = \frac{50 \times \sin 74^\circ}{\sin 44^\circ} = \frac{50 \times 0.9613}{0.6947} \approx 69.2\text{ m}.

No boat, no tape stretched across the water — just one measured line and two angles. That single idea, scaled up, is how whole countries were mapped.

Two traps snare almost everyone with the sine rule:

You need a complete opposite pair first. The sine rule only fires if you know a side and the angle across from it, plus one more measurement. If all you have is two sides and the angle squeezed between them, there is no pair — reach instead for the cosine rule.
Beware the ambiguous case. When you find an angle with \sin^{-1}, your calculator hands back one answer between 0^\circ and 90^\circ — but \sin\theta = \sin(180^\circ - \theta), so an obtuse angle can give the exact same sine. If \sin^{-1} returns 43^\circ, then 137^\circ may fit too — and the triangle could genuinely have two different shapes. Always ask: does the second angle still leave a sensible triangle (do all three angles add to under 180^\circ)? If both do, there are two valid answers; if only one does, keep that one.

For two thousand years, trigonometry was chained to the right angle. The sine and cosine rules snapped that chain — suddenly you could solve any triangle from just a few measurements. That unlocked triangulation: measure one baseline carefully, sight the angles to a distant point, and the sine rule gives you every other distance for free. Chain triangle to triangle and you can march a survey across an entire continent — which is exactly how the great 18th- and 19th-century surveys mapped India, France and Britain to astonishing accuracy, long before satellites.

The same idea steers ships and lands aircraft — and it's hiding in your pocket right now. GPS works by solving triangles between your phone and the satellites overhead: from the tiny time delays it works out distances, then triangulates a fix on where you're standing, to within a few metres. Every time a map app says "you are here," a triangle just got solved.