Exact values of inverse trig functions
Problem 13.62 · easy
Find the exact value of \( \displaystyle \arctan{\left(\sqrt{3} \right)} \).
- arctan returns the one angle in (−π/2, π/2) whose tan is sqrt(3).Reviewed
- \[ \tan{\left(\frac{\pi}{3} \right)} = \sqrt{3} \]tan(pi/3) = sqrt(3), and pi/3 is in (−π/2, π/2).✓ Proved
- \[ \operatorname{atan}{\left(\sqrt{3} \right)} = \frac{\pi}{3} \]So that angle is the value.✓ Proved
Answer \( \frac{\pi}{3} \)
✓ 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's inverse function |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the range of the arctan function and verifies that pi/3 lies within this range and satisfies the tangent equation.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.