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Solving trigonometric equations

Problem 13.402 · medium

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

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

Answer \( \theta = 0,\ \pi \)

Lines: 3 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
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 is incomplete because it fails to identify theta = 3pi/2, which is a valid solution in the interval [0, 2pi) corresponding to cos(theta) = -1.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-21 — The solution is incomplete because it fails to identify theta = 3pi/2, which is a valid solution in the interval [0, 2pi) corresponding to cos(theta) = -1.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution is incomplete because it fails to solve for the case cos(theta) = -1, which yields theta = pi. While pi is listed in the final answer, the step 'Set each factor to zero' is not followed by finding the solutions for the second factor (cos(theta) + 1 = 0). The verification steps only check theta = 0 and theta = pi, but do not derive theta = pi from the equation cos(theta) = -1. A student might miss that theta = pi comes from the negative factor, or might think only theta = 0 is a solution if they ignore the second factor. The solution skips the actual solving process for half the roots.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution is incomplete because it fails to identify theta = pi/2 and theta = 3pi/2, which are solutions corresponding to cos(theta) = -1. The stated answer is missing half of the valid solutions.
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.