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Solving trigonometric equations

Problem 13.408 · easy

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

Isolate cos θ and find the angles in [0, 2π) with that value.
\[ \left. 2 \cos{\left(\theta \right)} - 1 \right|_{\substack{ \theta=\frac{\pi}{3} }} = 0 \]
θ = pi/3 works.✓ Proved
\[ \left. 2 \cos{\left(\theta \right)} - 1 \right|_{\substack{ \theta=\frac{5 \pi}{3} }} = 0 \]
θ = 5*pi/3 works.✓ Proved

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

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 verifies that the given answers are solutions but fails to demonstrate that they are the only solutions in the specified interval. It omits the logical step of determining all angles with the given cosine value, leaving open the possibility of missing solutions.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-21 — The solution verifies that the given answers are solutions but fails to demonstrate that they are the only solutions in the specified interval. It omits the logical step of determining all angles with the given cosine value, leaving open the possibility of missing solutions.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: fail (misleading) 2026-10-20 — The solution only verifies that the given answers are solutions, but fails to demonstrate that they are the *only* solutions in the interval [0, 2π). A student would learn that checking candidates is sufficient for solving an equation, which is false.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution fails to show the derivation of the angles, merely verifying the final answers. It does not explain why these are the only solutions in the given interval, omitting the crucial step of identifying the reference angle and the quadrants where cosine is positive.
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.