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Exact values of inverse trig functions

Problem 13.66 · easy

Find the exact value of \( \displaystyle \arctan{\left(\frac{\sqrt{3}}{3} \right)} \).

arctan returns the one angle in (−π/2, π/2) whose tan is sqrt(3)/3.
Reviewed
\[ \tan{\left(\frac{\pi}{6} \right)} = \frac{\sqrt{3}}{3} \]
tan(pi/6) = sqrt(3)/3, and pi/6 is in (−π/2, π/2).✓ Proved
\[ \operatorname{atan}{\left(\frac{\sqrt{3}}{3} \right)} = \frac{\pi}{6} \]
So that angle is the value.✓ Proved

Answer \( \frac{\pi}{6} \)

✓ 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's inverse function

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the range of the arctangent function and verifies that the proposed angle lies within this range and satisfies the tangent equation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the range of the arctangent function and verifies that the proposed angle lies within this range and satisfies the tangent equation.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the range of the arctangent function and verifies that pi/6 is the unique angle in that range with the specified tangent value.
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/inverse_trig_values, checked 2026-10-17 with SymPy 1.14.0.