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Solving trigonometric equations

Problem 13.86 · easy

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

Isolate tan θ and find the angles in [0, 2π) with that value.
\[ \left. \tan{\left(\theta \right)} - \frac{\sqrt{3}}{3} \right|_{\substack{ \theta=\frac{\pi}{6} }} = 0 \]
θ = pi/6 works.✓ Proved
\[ \left. \tan{\left(\theta \right)} - \frac{\sqrt{3}}{3} \right|_{\substack{ \theta=\frac{7 \pi}{6} }} = 0 \]
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 verifies that the proposed answers are solutions but fails to prove they are the only solutions in the given interval. It does not use the periodicity of the tangent function or the unit circle to derive the general solution and restrict it to [0, 2π).

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution verifies that the proposed answers are solutions but fails to prove they are the only solutions in the given interval. It does not use the periodicity of the tangent function or the unit circle to derive the general solution and restrict it to [0, 2π).
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution verifies that the stated answers are solutions but fails to demonstrate that they are the *only* solutions in the given interval. A student might incorrectly conclude that checking specific values is sufficient to solve an equation, rather than using the periodicity of the tangent function to find the general solution and then restricting it to the domain.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the reference angle and the quadrants where tangent is positive, providing the correct general solutions within the specified interval.
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.