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Solving trigonometric equations

Problem 13.482 · easy

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

Isolate sin θ and find the angles in [0, 2π) with that value.
\[ \left. 2 \sin{\left(\theta \right)} + \sqrt{3} \right|_{\substack{ \theta=\frac{4 \pi}{3} }} = 0 \]
θ = 4*pi/3 works.✓ Proved
\[ \left. 2 \sin{\left(\theta \right)} + \sqrt{3} \right|_{\substack{ \theta=\frac{5 \pi}{3} }} = 0 \]
θ = 5*pi/3 works.✓ Proved

Answer \( \theta = \frac{4 \pi}{3},\ \frac{5 \pi}{3} \)

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 the two specific answers but fails to demonstrate that these are the only solutions in the interval [0, 2π). It omits the derivation (e.g., using the unit circle or reference angles) required to justify the completeness of the solution set.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-09 — The solution only verifies the two specific answers but fails to demonstrate that these are the only solutions in the interval [0, 2π). It omits the derivation (e.g., using the unit circle or reference angles) required to justify the completeness of the solution set.
gpt-oss:20b: pass 2026-11-09
qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution claims to isolate sin(theta) and find the angles, but it skips the derivation entirely and only verifies the final answers. This fails to demonstrate the method for solving the equation, specifically determining the reference angle and identifying the correct quadrants (III and IV) where sine is negative.
gpt-oss:20b: pass 2026-11-08
qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly isolates the sine function, identifies the correct quadrants (III and IV) for the negative value, and verifies the solutions within the specified domain.
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.