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Solving trigonometric equations

Problem 13.85 · easy

Solve \( \displaystyle \cos{\left(\theta \right)} = - \frac{1}{2} \) for \( \displaystyle 0 \le \theta < 2\pi \).

Isolate cos θ and find the angles in [0, 2π) with that value.
\[ \left. \cos{\left(\theta \right)} + 1 \cdot \frac{1}{2} \right|_{\substack{ \theta=\frac{2 \pi}{3} }} = 0 \]
θ = 2*pi/3 works.✓ Proved
\[ \left. \cos{\left(\theta \right)} + 1 \cdot \frac{1}{2} \right|_{\substack{ \theta=\frac{4 \pi}{3} }} = 0 \]
θ = 4*pi/3 works.✓ Proved

Answer \( \theta = \frac{2 \pi}{3},\ \frac{4 \pi}{3} \)

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 correct but fails to demonstrate that they are the only solutions in the given interval. It omits the logical step of identifying the reference angle and determining the quadrants where cosine is negative.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution only verifies that the proposed answers are correct but fails to demonstrate that they are the only solutions in the given interval. It omits the logical step of identifying the reference angle and determining the quadrants where cosine is negative.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution only verifies that the proposed answers are correct but fails to demonstrate that they are the only solutions in the given interval. It lacks the logical step of identifying the reference angle and determining the quadrants where cosine is negative.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to derive the answers, merely verifying them after the fact. It omits the necessary step of identifying the reference angle and determining the correct quadrants (II and III) where cosine is negative.
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.