PlaySpectrum

Helper · book companion path

Prior: When the circle keeps turning · Next: When the turn uses a rope length · Shelf: Kurs 4

Math / Helper companion / Tangent

Uncut · book companion · ~54–56

Climb from sin/cos — not a radians side note

Height ÷ run

Score 0 / 100

Sin and cos give height and sideways. Sometimes the job asks for their ratio — steepness, slope, or how fast a scan climbs between walls. Climb from height÷run on the tip, then read the walls where run hits zero. This is its own rung (~54–56), not a fake bridge inside radians. (tangens · asymptot · period)

Vortex

V1 — Height ÷ run on the tip.

Builds on: sin = height, cos = sideways (trig functions / unit circle).

Tool gained: tan θ = sin θ / cos θ (= opp/adj in the acute case).

  1. Principle. Steepness is height compared with run — the ratio of the two tip coordinates you already have.
  2. Model. tan θ = sin θ / cos θ. On a right triangle, that is still opp/adj.
  3. Prediction / Answer. A ramp rises 3 m over 4 m run → tan θ = 3/4. Same ratio for a scan elevation angle.
Vortex

V2 — Walls where run vanishes.

Builds on: tan = sin/cos (V1 tool).

Tool gained: vertical asymptotes where cos = 0; period of tan(kx) is 180°/k.

  1. Principle. When sideways (cos) is zero, the ratio shoots to ±∞ — walls the graph never crosses. Between neighbouring walls the climb repeats once.
  2. Model. Asymptotes where cos(kx) = 0. For y = tan(kx), P = 180°/k.
  3. Prediction / Answer. tan(3x) has P = 60°. Space safety cutouts between walls — same P tool with a machine that can jam.

1. Poor dump → rewrite

Poor explanation

Book move: tuck tangent under radians as a side formula. Our rewrite: tan earns its own climb from sin/cos; radians can come after for models that need rope measure — Builds on this tool, not a fake merge.

Tool · weapon

Weapons: tan = sin/cos → walls at cos = 0 → P = 180°/k for tan(kx).

2. Hands-on — steepness and walls

Drag θ near 90°. Watch tan blow up as cos → 0.

…

Fake → real

Fake

“State the period of tan(3x).”

Real

Lidar scan ∝ tan(3x). P = 60°. Designers space cutouts between walls — same formula, jam risk if ignored.

Gate — four, all required

1. Reconstruct tan from prior tools.

2. Why does tan blow up at 90°?

3. Predict P for y = tan(2x).

4. Climb graph: tangent vs radians-models.