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Solving trigonometric equations

Problem 13.84 · easy

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

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

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — 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 show the derivation or argument for uniqueness.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution verifies that the given answers are roots but fails to demonstrate that they are the *only* solutions in the specified interval. It does not show the derivation (e.g., using the unit circle or reference angles) to ensure completeness.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution is incomplete because it only verifies the proposed answers rather than deriving them. It fails to demonstrate how the values pi/6 and 5*pi/6 were obtained from sin(theta) = 1/2, omitting the crucial step of identifying the reference angle and the quadrants where sine is positive.
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/trig_equations, checked 2026-10-17 with SymPy 1.14.0.