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Solving trigonometric equations

Problem 13.252 · easy

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

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

Answer \( \theta = \frac{2 \pi}{3},\ \frac{5 \pi}{3} \)

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 that the proposed answers are solutions, but fails to demonstrate that these are the *only* solutions in the given interval. It lacks the logical step of identifying the reference angle and determining the quadrants where tangent is negative.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution only verifies that the proposed answers are solutions, but fails to demonstrate that these are the *only* solutions in the given interval. It lacks the logical step of identifying the reference angle and determining the quadrants where tangent is negative.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to demonstrate how the specific angles were derived, merely verifying them. It omits the crucial step of identifying the reference angle and determining the correct quadrants (II and IV) where tangent is negative.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the quadrants where tangent is negative and verifies the specific angles. The setup and conclusion are sound.
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.