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Solving trigonometric equations

Problem 13.263 · 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.
\[ \left. 2 \tan{\left(\theta \right)} - \frac{2 \sqrt{3}}{3} \right|_{\substack{ \theta=\frac{\pi}{6} }} = 0 \]
θ = pi/6 works.✓ Proved
\[ \left. 2 \tan{\left(\theta \right)} - \frac{2 \sqrt{3}}{3} \right|_{\substack{ \theta=\frac{7 \pi}{6} }} = 0 \]
θ = 7*pi/6 works.✓ Proved

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

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 stated answers are solutions, but it fails to prove that these are the *only* solutions in the given interval. A complete solution must derive the general solution or explicitly check all quadrants to ensure no other angles satisfy the equation.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution only verifies that the stated answers are solutions, but it fails to prove that these are the *only* solutions in the given interval. A complete solution must derive the general solution or explicitly check all quadrants to ensure no other angles satisfy the equation.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution only verifies that the given answers are solutions, but it fails to demonstrate that they are the *only* solutions in the interval [0, 2π). A complete solution must derive the general solution or explicitly check all quadrants to ensure no other solutions exist.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the reference angle and applies the periodicity of the tangent function to find all solutions within the specified interval [0, 2π). The verification steps confirm the results.
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.