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Solving trigonometric equations

Problem 13.406 · easy

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

Isolate tan θ and find the angles in [0, 2π) with that value.
Reviewed
\[ \left. 2 \tan{\left(\theta \right)} + \frac{2 \sqrt{3}}{3} \right|_{\substack{ \theta=\frac{5 \pi}{6} }} = 0 \]
θ = 5*pi/6 works.✓ Proved
\[ \left. 2 \tan{\left(\theta \right)} + \frac{2 \sqrt{3}}{3} \right|_{\substack{ \theta=\frac{11 \pi}{6} }} = 0 \]
θ = 11*pi/6 works.✓ Proved

Answer \( \theta = \frac{5 \pi}{6},\ \frac{11 \pi}{6} \)

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 the quadrants where tangent is negative, providing the correct solutions within the specified interval.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly identifies the reference angle and the quadrants where tangent is negative, providing the correct solutions within the specified interval.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution verifies that the stated answers are solutions but fails to demonstrate that they are the *only* solutions in the given interval. A complete solution must derive the general solution or explicitly check all quadrants to ensure no roots are missed.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution claims to isolate tan(theta) and find angles, but provides no derivation or justification for the specific values 5pi/6 and 11pi/6. It merely verifies the answers without showing the logical steps (finding the reference angle and determining quadrants) required to solve the problem.
gpt-oss:20b: pass 2026-10-20

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 2026-10-21 with SymPy 1.14.0.