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Solving trigonometric equations

Problem 13.322 · 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.
\[ \left. 2 \cos^{2}{\left(\theta \right)} - \cos{\left(\theta \right)} - 1 \right|_{\substack{ \theta=0 }} = 0 \]
θ = 0 works.✓ Proved
\[ \left. 2 \cos^{2}{\left(\theta \right)} - \cos{\left(\theta \right)} - 1 \right|_{\substack{ \theta=\frac{2 \pi}{3} }} = 0 \]
θ = 2*pi/3 works.✓ Proved
\[ \left. 2 \cos^{2}{\left(\theta \right)} - \cos{\left(\theta \right)} - 1 \right|_{\substack{ \theta=\frac{4 \pi}{3} }} = 0 \]
θ = 4*pi/3 works.✓ Proved

Answer \( \theta = 0,\ \frac{2 \pi}{3},\ \frac{4 \pi}{3} \)

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 interval [0, 2π). Specifically, it omits the derivation of θ = 2π from cos(θ) = 1, which is a valid solution in the given range.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution verifies that the stated answers are roots but fails to demonstrate that they are the *only* solutions in the interval [0, 2π). Specifically, it omits the derivation of θ = 2π from cos(θ) = 1, which is a valid solution in the given range.
gpt-oss:20b: pass 2026-10-19
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. It omits the step of solving cos(θ) = 1 and cos(θ) = -1/2 to derive the angles, leaving the completeness of the solution unjustified.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution is incomplete because it fails to identify the solution corresponding to cos(theta) = 1 within the interval [0, 2pi). While theta = 0 is a solution, the factor (cos(theta) - 1) = 0 implies cos(theta) = 1, which also yields theta = 0 (already listed) but the logic presented skips the explicit derivation of the roots from the factors, merely checking specific values. More critically, the factorization yields cos(theta) = 1 and cos(theta) = -1/2. The solutions for cos(theta) = -1/2 are 2pi/3 and 4pi/3. The solution for cos(theta) = 1 is 0. The stated answer is correct, but the solution steps do not actually *solve* for the angles; they only verify three specific points. A student reading this would not learn how to find the angles from the cosine values. However, looking closer at the prompt's constraints: 'find what is wrong'. The steps provided are: 1. Factor. 2. Set factors to zero. 3-5. Verify specific answers. The step 'Set each factor to zero' is not followed by the actual solving of cos(theta)=1 or cos(theta)=-1/2. It jumps to verification. This is a logical gap. But is it an 'error' in the solution provided? The solution claims to solve it. It lists the correct answers. But the *method* shown is just verification. If I must be adversarial: The solution does not demonstrate how the answers were derived, only that they are correct. This is misleading as a 'solution'. However, usually 'error' implies a wrong result or invalid step. The steps are valid (verification is valid). But it's not a complete solution. Let's look for a harder error. Did it miss any solutions? cos(theta)=1 -> theta=0. cos(theta)=-1/2 -> theta=2pi/3, 4pi/3. All solutions in [0, 2pi) are found. The set of solutions is correct. The issue is purely pedagogical/methodological: it verifies rather than derives. Is that 'misleading' or 'style'? It teaches that checking answers is solving. That is misleading. But wait, look at step 1. The factorization is correct. Step 2 says 'Set each factor to zero; then find every angle...'. Steps 3-5 check specific angles. It never actually *finds* them. It assumes them. This is a 'fail' because it doesn't actually solve the problem, it just checks the answer key. A student would learn to guess and check. That is misleading.
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 2026-10-19 with SymPy 1.14.0.