Exact trigonometric values
Problem 13.235 · easy
Convert \( \displaystyle 240^\circ \) to radians, as a multiple of \( \displaystyle \pi \).
- Multiply by π/180, since 180° = π radians.Reviewed
- \[ \frac{240 \pi}{180} = \frac{4 \pi}{3} \]Simplify the fraction.✓ Proved
Answer \( \frac{4 \pi}{3} \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | deg·π/180 in floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the conversion factor and simplifies the fraction accurately.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the conversion factor and simplifies the fraction accurately.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies the conversion factor and simplifies the fraction accurately.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The conversion factor and arithmetic are correct.gpt-oss:20b: pass 2026-10-15
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/trig_values, checked 2026-10-18 with SymPy 1.14.0.