Exact values of inverse trig functions
Problem 13.69 · easy
Find the exact value of \( \displaystyle \arccos{\left(- \frac{1}{2} \right)} \).
- arccos returns the one angle in [0, π] whose cos is -1/2.Reviewed
- \[ \cos{\left(\frac{2 \pi}{3} \right)} = - \frac{1}{2} \]cos(2*pi/3) = -1/2, and 2*pi/3 is in [0, π].✓ Proved
- \[ \operatorname{acos}{\left(- \frac{1}{2} \right)} = \frac{2 \pi}{3} \]So that angle is the value.✓ Proved
Answer \( \frac{2 \pi}{3} \)
✓ 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 2026-10-14 — The solution correctly identifies the range of the arccos function and verifies that the proposed answer lies within that range and satisfies the cosine equation.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the range of the arccos function and verifies that the proposed angle lies within this range and satisfies the cosine 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.