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Exact values of inverse trig functions

Problem 13.65 · easy

Find the exact value of \( \displaystyle \arccos{\left(-1 \right)} \).

arccos returns the one angle in [0, π] whose cos is -1.
Reviewed
\[ \cos{\left(\pi \right)} = -1 \]
cos(pi) = -1, and pi is in [0, π].✓ Proved
\[ \operatorname{acos}{\left(-1 \right)} = \pi \]
So that angle is the value.✓ Proved

Answer \( \pi \)

✓ 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the range of the arccos function and verifies that pi is the unique value in that range satisfying the condition.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the range of the arccos function and the specific angle within that range whose cosine is -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 2026-10-17 with SymPy 1.14.0.