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Solving trigonometric equations

Problem 13.485 · easy

Solve \( \displaystyle \cos{\left(\theta \right)} = \frac{\sqrt{3}}{2} \) for \( \displaystyle 0 \le \theta < 2\pi \).

Isolate cos θ and find the angles in [0, 2π) with that value.
Reviewed
\[ \left. \cos{\left(\theta \right)} - \frac{\sqrt{3}}{2} \right|_{\substack{ \theta=\frac{\pi}{6} }} = 0 \]
θ = pi/6 works.✓ Proved
\[ \left. \cos{\left(\theta \right)} - \frac{\sqrt{3}}{2} \right|_{\substack{ \theta=\frac{11 \pi}{6} }} = 0 \]
θ = 11*pi/6 works.✓ Proved

Answer \( \theta = \frac{\pi}{6},\ \frac{11 \pi}{6} \)

Lines: 2 proved, 1 reviewed. 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	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	Newton's method from 1440 starting points finds exactly 2 roots in [0, 2π), the same ones

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the two solutions in the specified interval and verifies them. The reasoning is sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-09 — The solution correctly identifies the two solutions in the specified interval and verifies them. The reasoning is sound.
gpt-oss:20b: pass 2026-11-09
qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution only verifies that the proposed answers are solutions, but it fails to prove that these are the *only* solutions in the given interval. A complete solution must argue that no other angles satisfy the equation (e.g., by referencing the unit circle or the periodicity of cosine).
gpt-oss:20b: pass 2026-11-08
qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution fails to explain how the solutions were derived, merely verifying them. It omits the crucial step of identifying the reference angle and determining the correct quadrants (I and IV) for positive cosine values.
gpt-oss:20b: pass 2026-11-08

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 2026-11-09 with SymPy 1.14.0.