Math / A length you cannot reach
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One length and an angle
Almost everything we measure in the world is made of circles, triangles, and rectangles (say rectangle when the sides need not be equal). A triangle is special: three lengths and three angles lock each other. Once enough pieces are known, the rest are forced. That lock is the power. Names — , cosine, — are just labels for the three ways to pair sides with an angle. The meaning is the lock (trigonometri i rätvinklig triangel).
A shadow as long as the man: then the pyramid’s shadow is as long as the pyramid. Parallel sun, same shape. Honesty: the story is late — Diogenes and Plutarch, centuries after. The method still works.
Noon at Syene, no shadow in a well. Same noon at Alexandria, a stick shows the sun 1/50 of a turn off upright. Distance between the towns times 50: a number for the whole Earth. Honesty: we do not know his stadion exactly, so the “how close” argument never quite lands.
Sky triangles you cannot walk. He wrote chords: for each angle, the straight cut across a standard circle. Measure once, use forever. That is the table idea. The word trigonometry is much younger (Pitiscus, 1595).
Half-chord instead of whole chord — our . The name is a copy accident: Sanskrit jyā (bowstring) to Arabic without vowels, then Latin sinus (fold, bay). Your calculator still carries that mistranslation (sinus).
A stick of height 1, the length of its shadow at each sun angle. That table is , still called shadow (umbra) in the Latin books. Fincke named “tangent” in 1583 (tangens).
Hadley’s octant, then the sextant: a sailor reads a small angle from horizon to lamp or sun. Same ratio tables, now at sea.
On the public river this tool comes after similar triangles and the third-side rule. Those pages are not written yet, so this stop stays open — a length you cannot tape, not a side track.
You arrive at a river. You need the width — are your bridge materials enough? You cannot swim a tape across. Walk a baseline along your bank. Sight the tree on the far bank. Read the angle between bank and sight-line. Width = baseline × tan(angle). Drag both. Watch the width update. No abstract notation first — the job is the width (bredd = bas · tangens).
Right angle where the tree’s “straight across” meets your bank. Baseline is next to your angle (adjacent). Width is across from your angle (opposite). Their quotient belongs to the angle — that is .
Baseline 80 m · angle 50° · width ≈ 95 m
Your bank at the bottom. Water in the middle. Far bank and tree at the top. Drag baseline and angle; the width is forced · tan θ = motstående / närliggande.
Samma vinkel, samma form. Förhållandet hör till vinkeln. Sinus parar motstående med hypotenusan; cosinus parar närliggande med hypotenusan.
In the way: water, a drop, a wall. The square-on-the-sides rule wants two lengths. You have one side and an angle. That rule is silent. The angle’s ratio speaks.
Cover the number. Say a range. Then open.
1. Mast and ladder. You stand 24 m from a mast. The lamp sits 38° up from your eye. Do you need a 20 m ladder, or a 12 m one, once you add your eye height (1.6 m)?
Estimate: 38° is a bit less than half a right angle, so height above eye a bit less than 24 — maybe around 18.
Height above eye = 24 × tan 38° ≈ 24 × 0.781 = 18.7 m. Plus 1.6 m ≈ 20.3 m. The short ladder is a lie.
2. Stick and flagpole. A 1.2 m stick throws a 0.8 m shadow. At the same minute a flagpole’s shadow is 14 m. How tall is the pole?
Estimate: the stick is taller than its shadow, so the pole is taller than 14 — maybe around 20.
Same sun, same angle, same ratio. 1.2 / 0.8 = 1.5. Pole = 14 × 1.5 = 21 m. You did not need the word tangent. You used the table idea on two similar right triangles.
3. Guy line. A 9 m wire from the ground to a mast-top, 32° up from the ground. How far out is the peg, and how high is the top?
Estimate: 32° is a shallow slope, so the peg is almost the full 9 m out, height a good bit less than 9.
The wire is the long side. Peg (adjacent) = 9 × cos 32° ≈ 9 × 0.848 = 7.6 m. Height (opposite) = 9 × sin 32° ≈ 9 × 0.530 = 4.8 m. Tangent would need a ground length you do not have yet.
Each right gate item is 20 points. Hands-on (drag baseline or angle on the river) is 20. Wrong is 0 until you retry and get it. Next unlocks only at 100%.
1. What is this tool doing?
2. You drag the angle on the river. The width / baseline number holds still for that angle, even if the drawing scales. Why?
3. You walk 40 m along the bank. The far tree sits 30° off the bank. Width of the water?
4. Which is a real question?