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Solving trigonometric equations

Problem 13.81 · easy

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

Isolate cos θ and find the angles in [0, 2π) with that value.
\[ \left. 2 \cos{\left(\theta \right)} - \sqrt{3} \right|_{\substack{ \theta=\frac{\pi}{6} }} = 0 \]
θ = pi/6 works.✓ Proved
\[ \left. 2 \cos{\left(\theta \right)} - \sqrt{3} \right|_{\substack{ \theta=\frac{11 \pi}{6} }} = 0 \]
θ = 11*pi/6 works.✓ Proved

Answer \( \theta = \frac{\pi}{6},\ \frac{11 \pi}{6} \)

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 stated answers are solutions, but it fails to demonstrate that these are the *only* solutions in the interval [0, 2π). A complete solution requires deriving the values from cos(θ) = √3/2, not just checking them.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution verifies that the stated answers are solutions, but it fails to demonstrate that these are the *only* solutions in the interval [0, 2π). A complete solution requires deriving the values from cos(θ) = √3/2, not just checking them.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution only verifies that the given answers are solutions, but it does not demonstrate that these 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 roots are missed.
gpt-oss:20b: fail (error) 2026-10-14 — The solution only verifies two candidate angles but does not justify that these are the only solutions in [0,2π). The initial sentence states to find all angles, yet the steps lack the necessary argument (e.g., solving cos θ=√3/2 and noting the period) to show completeness.
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution claims to isolate cos(theta) but fails to perform the division by 2, leaving the equation as 2cos(theta) = sqrt(3) implicitly. It then checks specific values without showing the derivation of the reference angle or the quadrant logic, which is the core of the problem.
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.