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Solving trigonometric equations

Problem 13.261 · medium

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

\[ 2 \cos^{2}{\left(\theta \right)} + \cos{\left(\theta \right)} - 1 = \left(\cos{\left(\theta \right)} + 1\right) \left(2 \cos{\left(\theta \right)} - 1\right) \]
Factor as a quadratic in cos θ.✓ Proved
Set each factor to zero; then find every angle in [0, 2π) with those values.
\[ \left. 2 \cos^{2}{\left(\theta \right)} + \cos{\left(\theta \right)} - 1 \right|_{\substack{ \theta=\frac{\pi}{3} }} = 0 \]
θ = pi/3 works.✓ Proved
\[ \left. 2 \cos^{2}{\left(\theta \right)} + \cos{\left(\theta \right)} - 1 \right|_{\substack{ \theta=\pi }} = 0 \]
θ = pi works.✓ Proved
\[ \left. 2 \cos^{2}{\left(\theta \right)} + \cos{\left(\theta \right)} - 1 \right|_{\substack{ \theta=\frac{5 \pi}{3} }} = 0 \]
θ = 5*pi/3 works.✓ Proved

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

Lines: 4 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	a sentence; read, not computed
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	✓ 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 3 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 roots but fails to demonstrate that they are the *only* solutions in the given interval. It omits the step of solving for cos(theta) = -1 and cos(theta) = 1/2 to derive the angles, leaving the completeness of the solution set unproven.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution verifies that the stated answers are roots but fails to demonstrate that they are the *only* solutions in the given interval. It omits the step of solving for cos(theta) = -1 and cos(theta) = 1/2 to derive the angles, leaving the completeness of the solution set unproven.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution verifies that the stated answers are roots but fails to demonstrate that they are the *only* solutions in the interval [0, 2π). It omits the crucial step of solving cos(θ) = -1 and cos(θ) = 1/2 to derive the specific angle values, leaving the completeness of the solution set unproven.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly factors the quadratic equation, identifies the valid cosine values, and verifies the resulting angles within the specified domain. The logic is sound and complete.
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.