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Solving trigonometric equations

Problem 13.407 · 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.
\[ \left. \cos{\left(\theta \right)} - 1 \right|_{\substack{ \theta=0 }} = 0 \]
θ = 0 works.✓ Proved

Answer \( \theta = 0 \)

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 = 0 is a solution but fails to establish that it is the unique solution in the interval [0, 2pi). It does not check other potential candidates (like 2pi, which is excluded, or argue via the range of cosine) to prove completeness.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-21 — The solution only verifies that theta = 0 is a solution but fails to establish that it is the unique solution in the interval [0, 2pi). It does not check other potential candidates (like 2pi, which is excluded, or argue via the range of cosine) to prove completeness.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution only verifies that theta = 0 is a solution but fails to prove it is the unique solution in the given interval. It does not address other potential solutions or explain why no other angles satisfy the equation.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution fails to explicitly state that theta = 0 is the unique solution in the interval [0, 2pi). Merely checking that theta = 0 works does not prove that no other solutions exist, which is a critical part of solving trigonometric equations.
gpt-oss:20b: fail (misleading) 2026-10-20 — The solution only checks θ=0 by substitution and does not justify that it is the only solution in [0,2π). It fails to address the possibility of θ=2π (which is excluded) or to explain why no other angles satisfy cosθ=1. This could mislead a student into thinking that checking a single value suffices.

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 2026-10-21 with SymPy 1.14.0.