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Solving trigonometric equations

Problem 13.259 · medium

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

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

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

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 only verifies that the stated answers are solutions, but it fails to demonstrate that these are the *only* solutions in the interval. Specifically, it omits the derivation of the solutions from the factors (cos(theta)=1 and cos(theta)=0) and does not check for other potential solutions like theta=2pi (though excluded by strict inequality, the logic must be explicit) or ensure no other angles satisfy cos(theta)=1 within the range.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution only verifies that the stated answers are solutions, but it fails to demonstrate that these are the *only* solutions in the interval. Specifically, it omits the derivation of the solutions from the factors (cos(theta)=1 and cos(theta)=0) and does not check for other potential solutions like theta=2pi (though excluded by strict inequality, the logic must be explicit) or ensure no other angles satisfy cos(theta)=1 within the range.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to solve the factored equation for all roots. It ignores the case cos(theta) = 1, which yields theta = 0, but implicitly treats it as a separate check rather than solving the system. More critically, it does not explicitly derive theta = 0 from cos(theta)=1, nor does it explain why no other solutions exist for cos(theta)=0 or cos(theta)=1 in the given interval, relying instead on verification of specific points without showing completeness.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly solve the factored equation for cos(theta), skipping the derivation of the candidate values (cos(theta)=0 and cos(theta)=1). It also omits the solution theta = 2pi is excluded but theta=0 is included, yet it does not explain why theta=pi is not a solution (cos(pi)=-1, not 0 or 1) or how the specific values were chosen other than checking them. More critically, it presents a verification of three points as the solution method, rather than deriving the general solution set. A student might think checking random points is sufficient. However, the most glaring issue is that it doesn't show the step: cos(theta)=0 => theta=pi/2, 3pi/2 and cos(theta)=1 => theta=0. It just jumps to checking specific values. This is a 'fail' because the logical gap is too large for a 'style' issue; it's an incomplete solution.
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.