Exact values of inverse trig functions
Problem 13.177 · easy
Find the exact value of \( \displaystyle \arccos{\left(1 \right)} \).
- arccos returns the one angle in [0, π] whose cos is 1.Reviewed
- \[ \cos{\left(0 \right)} = 1 \]cos(0) = 1, and 0 is in [0, π].✓ Proved
- \[ \operatorname{acos}{\left(1 \right)} = 0 \]So that angle is the value.✓ Proved
Answer \( 0 \)
✓ 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 2026-10-18gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the range of the arccosine function and verifies that 0 is the unique angle in that range with a cosine of 1.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the range of the arccos function and verifies that 0 is the unique angle in that range with a cosine of 1.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 2026-10-18 with SymPy 1.14.0.