At the boards, a triangle arrived as two sides and the angle wedged between them — and The Sine Law could not start, because no known side faced a known angle. The tool for exactly this arrangement looks like an old friend with a correction bolted on:
Pythagoras, corrected
Set and : the correction term vanishes and the law is the Pythagorean theorem. For angles under 90° the cosine is positive and the correction subtracts — the side facing a sharp angle is shorter than Pythagoras would guess. Past 90° the cosine goes negative, the subtraction becomes addition, and the facing side grows. The law is Pythagoras that knows what angle it is looking at.
With m, m, and between them:
Given all three sides instead, rearrange to hunt an angle:
Hunt the largest angle first — it faces the longest side — and a negative cosine on the way out is not an error; it is the law reporting an obtuse angle, exactly as Special Angles said it would.
Choosing your tool
One question decides
Does some known side face a known angle? Yes — sine law (or primary ratios, if a right angle is present). No — cosine law: two sides and the contained angle give a side; three sides give an angle. After the cosine law’s first move, a matched pair exists and the sine law reopens for the rest of the triangle.
Three-dimensional problems — a cliff across a river, a tower seen from two places — fall to the same two laws. The skill is spotting the pair of triangles that share an edge: solve the one you can, carry the shared edge to the other, and the third dimension never needs new mathematics. That carrying move is worth narrating out loud when you write solutions — it is the step Showing Your Thinking exists for.
The tool-choosing habit — and the triangles of How High Is That? — get their workout in Trig Ratios and Laws Practice.
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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