PUZZLE #4206: Euler's Formula (diff 3)

eulers-formula learn difficulty: 3/7 reason author: system unsolved

Euler's formula: e^(iθ) = cos θ + i sin θ. Each of the 7 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	`3.406046, 4.365699, 0.459197, 1.950916, 2.650956, 2.665042, -0.017051`
Word Length	`7`
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. 7 angles → 7 letters.`
Answer Format	`lowercase word, no spaces or punctuation (3-5 letters)`

{
  "type": "eulers-formula",
  "angles": [
    3.406046,
    4.365699,
    0.459197,
    1.950916,
    2.650956,
    2.665042,
    -0.017051
  ],
  "word_length": 7,
  "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. 7 angles \u2192 7 letters.",
  "answer_format": "lowercase word, no spaces or punctuation (3-5 letters)"
}

author's note: Pool fill: eulers-formula diff 3