Exact trigonometric values
Problem 13.48 · easy
Find the exact value of \( \displaystyle \csc{\left(\frac{\pi}{3} \right)} \).
- The angle pi/3 ends in quadrant 1 with reference angle pi/3, where csc is positive.Reviewed
- \[ \csc{\left(\frac{\pi}{3} \right)} = \frac{2 \sqrt{3}}{3} \]csc of the reference angle.✓ Proved
- \[ \csc{\left(\frac{\pi}{3} \right)} = \frac{2 \sqrt{3}}{3} \]Attach the sign for the quadrant.✓ Proved
Answer \( \frac{2 \sqrt{3}}{3} \)
✓ Nihil obstat Lines: 2 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 |
| 3 | ✓ 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 | mpmath evaluates the function at the angle |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution is correct and the reasoning is sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution is correct and the reasoning is sound.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution is correct and logically sound. The steps correctly identify the quadrant, reference angle, and sign, leading to the right value.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution is correct and the reasoning is sound.gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.