PUZZLE #4197: Euler's Formula (diff 4)

eulers-formula learn difficulty: 4/7 reason author: system unsolved

Euler's formula: e^(iθ) = cos θ + i sin θ. Each of the 8 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	`4.081402, 0.966562, 0.019988, 2.683464, 3.623714, -0.02737, 4.10708, 4.585612`
Word Length	`8`
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. 8 angles → 8 letters.`
Answer Format	`lowercase word, no spaces or punctuation (3-5 letters)`

{
  "type": "eulers-formula",
  "angles": [
    4.081402,
    0.966562,
    0.019988,
    2.683464,
    3.623714,
    -0.02737,
    4.10708,
    4.585612
  ],
  "word_length": 8,
  "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. 8 angles \u2192 8 letters.",
  "answer_format": "lowercase word, no spaces or punctuation (3-5 letters)"
}

author's note: Pool fill: eulers-formula diff 4