The Sine Law

The surveying triangles at the boards had no right angle anywhere, and the primary ratios from Grade 10 refused to start — opposite and hypotenuse mean nothing without a hypotenuse. Oblique triangles need their own tools. Label a triangle the standard way — side $a$ facing angle $A$, side $b$ facing $B$, side $c$ facing $C$ — and:

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

Every side, divided by the sine of the angle it faces, gives the same number. Drag a vertex around in Desmos or any dynamic geometry tool and watch the three fractions move in lockstep; that hidden constant is the whole law.

When it applies

Each equation in the chain has four parts, so you can solve one as soon as three are known. That means the sine law needs a matched pair — a side together with the angle facing it — plus one more angle or side. Two angles and any side works (the third angle comes free from the 180° sum). A matched pair plus one extra side works too. But two sides with only the angle between them known gives no complete fraction to stand on — that arrangement belongs to The Cosine Law. Before either law, always ask: does some known side face a known angle?

In practice: with $\angle A=44°$, $\angle B=71°$, and $a=12$ m,

$$b=\frac{12\sin 71°}{\sin 44°}\approx 16.3\text{ m}$$

— and since $71°>44°$, the answer had to beat 12 m. Run that bigger-angle-faces-bigger-side comparison before every calculation; it costs one second and catches flipped fractions instantly.

The ambiguous case

Two triangles can hide in one question

Solving for an angle with the sine law can return a disguised pair. Given $\angle A=35°$, $a=7$, and $b=10$: $\sin B=\frac{10\sin 35°}{7}\approx 0.8194$, and as Special Angles showed, two angles under 180° share that sine — $B\approx 55°$ or $B\approx 125°$. Check both against the angle budget: $35°+55°$ and $35°+125°$ each stay under 180°, so both triangles genuinely exist, with different shapes and different third sides.

Picture side $a$ as a swinging arm hinged at $C$: too long to miss, it can touch the base on the near side of vertical or the far side. When a question hands you two sides and a non-included angle, name both candidates, test both, and report every triangle that survives. Finding the second triangle your groupmate missed — or the one you missed — is Mistakes Are Data in its natural habitat.

The sine law carried the measuring work in How High Is That?, and Trig Ratios and Laws Practice mixes it with the cosine law so that choosing becomes part of the skill.

Curriculum connection

D1.6

pose problems involving right triangles and oblique triangles in two-dimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law, and the sine law (including the ambiguous case)

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D1.7

pose problems involving right triangles and oblique triangles in three-dimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law, and the sine law Sample problem: Explain how a surveyor could find the height of a vertical cliff that is on the other side of a raging river, using a measuring tape, a theodolite, and some trigonometry. Determine what the surveyor might measure, and use hypothetical values for these data to calculate the height of the cliff.

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