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Solving trigonometric equations

Problem 13.329 · easy

Solve \( \displaystyle \cos{\left(\theta \right)} = -1 \) for \( \displaystyle 0 \le \theta < 2\pi \).

Isolate cos θ and find the angles in [0, 2π) with that value.
\[ \left. \cos{\left(\theta \right)} + 1 \right|_{\substack{ \theta=\pi }} = 0 \]
θ = pi works.✓ Proved

Answer \( \theta = \pi \)

Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
2	✓ 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	Newton's method from 1440 starting points finds exactly 1 roots in [0, 2π), the same ones

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution only verifies that theta = pi is a solution, but fails to prove it is the unique solution in the given interval. A complete solution requires arguing that pi is the only angle in [0, 2pi) where cosine equals -1.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution only verifies that theta = pi is a solution, but fails to prove it is the unique solution in the given interval. A complete solution requires arguing that pi is the only angle in [0, 2pi) where cosine equals -1.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to prove uniqueness. It only verifies that theta = pi is a solution, but does not demonstrate that it is the only solution in the interval [0, 2pi). A complete solution requires arguing that cosine is injective on [0, pi] or analyzing the unit circle to show no other angle satisfies the condition.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to establish uniqueness. It only verifies that theta = pi is a solution, but does not prove it is the only solution in the interval [0, 2pi). A complete solution must argue that cosine is injective on [0, pi] or use the unit circle to show no other angle satisfies the condition.
gpt-oss:20b: pass 2026-10-15

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/trig_equations, checked 2026-10-19 with SymPy 1.14.0.