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Exact trigonometric values

Problem 13.497 · easy

Find the exact value of \( \displaystyle \csc{\left(\frac{5 \pi}{3} \right)} \).

The angle 5*pi/3 ends in quadrant 4 with reference angle pi/3, where csc is negative.
Reviewed
\[ \csc{\left(\frac{\pi}{3} \right)} = \frac{2 \sqrt{3}}{3} \]
csc of the reference angle.✓ Proved
\[ \csc{\left(\frac{5 \pi}{3} \right)} = - \frac{2 \sqrt{3}}{3} \]
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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the quadrant, reference angle, and sign of the cosecant function. The steps are logically sound and lead to the correct exact value.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-10 — The solution correctly identifies the quadrant, reference angle, and sign of the cosecant function. The steps are logically sound and lead to the correct exact value.
gpt-oss:20b: pass 2026-11-10
qwen3.6:27b-mlx: pass 2026-11-09 — The solution correctly identifies the quadrant, reference angle, and sign for the cosecant function. The logic is sound and the final answer is correct.
gpt-oss:20b: pass 2026-11-09
qwen3.6:27b-mlx: pass 2026-11-09 — The solution correctly identifies the quadrant, reference angle, and sign for the cosecant function. The steps are logically sound and lead to the correct exact value.
gpt-oss:20b: pass 2026-11-09

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 2026-11-10 with SymPy 1.14.0.