Math / The span you cannot walk
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No right angle
You are on a sandbar in the middle. Both banks held — you cannot pace to either shore. Across the water sits a span you must judge: how long is that cable you cannot walk? Pace two lines you can walk on the island, and read the angle between them. No right corner. The three sides still lock — you need the cosine rule (two sides + included angle) or the sine rule (two angles + one side) (cosinussatsen / sinussatsen).
— that was the right-angle table. This page is when the corner is not square.
Land and sea jobs where you cannot stand under a right corner: a bay, an island, a reef. Pace what you can. Sight what you must. The triangle still closes.
Heron of Alexandria wrote area from three sides alone — no angle needed. The same spirit: enough pieces lock the rest. Honesty: the formula may be older; his book is what we still have.
Jamshīd al-Kāshī wrote the cosine form we still use: the third side from two sides and the angle between them. Surveyors and astronomers needed that when no corner was square (cosinussatsen).
Side over sine of opposite angle is the same for every corner of the triangle. Two angles and one side — or two sides and a non-included angle, carefully — unlock the rest (sinussatsen).
Honesty: we are not copying a commercial exercise sheet. The island is a job picture, not a scanned figure.
Drag side a, side b, and the included angle γ. The far side c updates by the cosine rule: c² = a² + b² − 2ab cos(γ). When γ is 90°, cos is 0 and you get the familiar square-on-the-sides rule back — a special case, not a different world (c² = a² + b² − 2ab cos γ).
a = 40 m · b = 55 m · γ = 70° · span c ≈ 56 m
You stand at the angle vertex on the sandbar. Sides a and b are lines you paced. Side c is the span across the water.
Cosinussatsen: c² = a² + b² − 2ab cos γ. Sinussatsen: a/sin A = b/sin B = c/sin C.
Pick the rule that matches what you can measure. On the island, pacing two lines and reading the included angle points to cosine. Sighting two angles from one end points to sine.
1. Cable across. From your sandbar you pace 30 m toward one pier mark and 45 m toward the other. The angle between those walks is 55°. How long is the cable between the marks?
Estimate: if the angle were 90°, c would be √(30²+45²)≈54. At 55° it is a bit less — maybe around 38–42?
c² = 30² + 45² − 2·30·45·cos 55° ≈ 900 + 2025 − 2700·0.574 ≈ 2925 − 1550 ≈ 1375. c ≈ √1375 ≈ 37 m. The estimate was a touch high; still in the right band.
2. Two angles. You know one side of a triangle is 20 m. The opposite angle is 40°. Another angle is 65°. What is the side opposite 65°?
Estimate: 65° is bigger than 40°, so its opposite side is bigger than 20 — maybe around 28–30.
Sine rule: a / sin A = b / sin B. So b = 20 · sin 65° / sin 40° ≈ 20 · 0.906 / 0.643 ≈ 28.2 m.
Hands-on (drag a side or the angle) is 20. Each right gate item is 20. Next unlocks at 100%.
1. What is this tool for?
2. You know two sides and the angle between them. Which rule finds the third side?
3. When the included angle is 90°, the cosine rule becomes…
4. Which is a real island job?