PUZZLE #3715: Euler's Formula (diff 7)

eulers-formula learn difficulty: 7/7 reason author: system unsolved

Euler's formula: e^(iθ) = cos θ + i sin θ. Each of the 12 angles below represents a point on the unit circle. Compute e^(iθ) for each, then find the landing position on the circle. The letter at position (angle × 26 / 2π) spells the answer word.

DATA

Angles	`0.702936, 1.906889, 1.238219, 1.211176, 0.985823, 4.082076, 0.961145, 3.119501, 4.613416, 1.9069, -0.007321, 2.629246`
Word Length	`12`
Hint	`For each angle θ, compute e^(iθ) = cos(θ) + i sin(θ). The landing position is atan2(sin θ, cos θ) which recovers the angle. letter_index = round(angle × 26 / (2 × π)) mod 26. 12 angles → 12 letters.`
Answer Format	`lowercase word, no spaces or punctuation (3-5 letters)`

{
  "type": "eulers-formula",
  "angles": [
    0.702936,
    1.906889,
    1.238219,
    1.211176,
    0.985823,
    4.082076,
    0.961145,
    3.119501,
    4.613416,
    1.9069,
    -0.007321,
    2.629246
  ],
  "word_length": 12,
  "hint": "For each angle \u03b8, compute e^(i\u03b8) = cos(\u03b8) + i sin(\u03b8). The landing position is atan2(sin \u03b8, cos \u03b8) which recovers the angle. letter_index = round(angle \u00d7 26 / (2 \u00d7 \u03c0)) mod 26. 12 angles \u2192 12 letters.",
  "answer_format": "lowercase word, no spaces or punctuation (3-5 letters)"
}

author's note: Pool fill: eulers-formula diff 7