Math / Two turns in a row
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Composing turns
Two turns in a row
You turn by α, then by β. The combined aim is α+β — turns add. Heights do not: sin(α+β) is not sin α + sin β. Why? Because height is a coordinate after the whole turn, not a pile of two separate heights. Expanding the combined turn into pieces of α and β is the addition picture (additionsformler) — purpose: aiming, music phase, or double-angle as the special case β = α.
Origins
Almagest
Chord addition for astronomy: combine two sky arcs. The modern sin(α+β) is the cleaned heir of that job.
phase
Two dials, two turns of a wheel, two phase shifts in a wave — you compose turns. Adding the height numbers alone aims wrong.
When the second turn equals the first, α+α = 2α. One special case of the same composition — not a separate religion.
Honesty: we show composition on the circle. We are not dumping every identity for its own sake.
Turn α, then β — watch the combined point
Drag α and β. The dashed arms show each turn. The solid arm is α+β. Read sin(α+β) against sin α + sin β — they disagree. The expansion cos α sin β + sin α cos β rebuilds the true combined height (sin(α+β) = sin α cos β + cos α sin β).
α=30° β=40° · sin(α+β)≈0.94 · sinα+sinβ≈0.64+0.64=1.28 (not equal)
Turns add on the rim. Coordinates do not add as a lazy sum — that is why the expansion exists.
Additionsformler: sin(α+β) = sin α cos β + cos α sin β. Cosinus liknande.
Why the lazy sum fails
Gate — four, all required
Hands-on (drag α or β) is 20. Each right gate item is 20.
1. What is this tool for?
2. Why is sin(α+β) not equal to sin α + sin β?
3. Double-angle (2α) is…
4. Which is a real job?