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Solving trigonometric equations

Problem 13.76 · 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.
Reviewed
\[ \left. 2 \sin{\left(\theta \right)} + 1 \right|_{\substack{ \theta=\frac{7 \pi}{6} }} = 0 \]
θ = 7*pi/6 works.✓ Proved
\[ \left. 2 \sin{\left(\theta \right)} + 1 \right|_{\substack{ \theta=\frac{11 \pi}{6} }} = 0 \]
θ = 11*pi/6 works.✓ Proved

Answer \( \theta = \frac{7 \pi}{6},\ \frac{11 \pi}{6} \)

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 reference angle and quadrants for sin(theta) = -1/2, and verifies both solutions within the specified domain.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the reference angle and quadrants for sin(theta) = -1/2, and verifies both solutions within the specified domain.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution verifies that the stated answers are correct but fails to demonstrate that they are the *only* solutions in the given interval. A complete solution must derive the values from the reference angle and quadrant analysis, rather than just checking candidates.
gpt-oss:20b: fail (misleading) 2026-10-14 — The solution only tests two angles without explaining why these are the only solutions. It omits the step of solving sin θ = −1/2 and showing that the two angles 7π/6 and 11π/6 are the only solutions in [0,2π).
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution claims to find angles but fails to explicitly derive or state the reference angle and quadrant logic required to justify the specific solutions. Merely checking the answers does not constitute solving the equation, and the sentence 'find the angles' is not a valid mathematical step without the intermediate reasoning.
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.