PUZZLE #3702: Euler's Formula (diff 5)

eulers-formula learn difficulty: 5/7 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	`0.9641, 1.942554, 3.154754, 4.337499, 4.598858, 0.947902, 1.960353, 3.128902`
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": [
    0.9641,
    1.942554,
    3.154754,
    4.337499,
    4.598858,
    0.947902,
    1.960353,
    3.128902
  ],
  "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 5