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Solving trigonometric equations

Problem 13.256 · easy

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

Isolate sin θ and find the angles in [0, 2π) with that value.
\[ \left. \sin{\left(\theta \right)} - 1 \cdot \frac{1}{2} \right|_{\substack{ \theta=\frac{\pi}{6} }} = 0 \]
θ = pi/6 works.✓ Proved
\[ \left. \sin{\left(\theta \right)} - 1 \cdot \frac{1}{2} \right|_{\substack{ \theta=\frac{5 \pi}{6} }} = 0 \]
θ = 5*pi/6 works.✓ Proved

Answer \( \theta = \frac{\pi}{6},\ \frac{5 \pi}{6} \)

Lines: 2 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
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: fail (error) — The solution only verifies that the given answers are correct but fails to demonstrate that they are the only solutions in the specified interval. It does not use the unit circle or reference angles to justify why no other solutions exist.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution only verifies that the given answers are correct but fails to demonstrate that they are the only solutions in the specified interval. It does not use the unit circle or reference angles to justify why no other solutions exist.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution is incomplete because it fails to state that these are the only solutions in the given interval. Verifying that two values work does not prove that no other values exist.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explain how the specific angles were derived (e.g., using the unit circle or reference angles). It merely verifies the answers, which does not constitute a solution method for a student who does not already know the answers.
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 2026-10-18 with SymPy 1.14.0.