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Solving trigonometric equations

Problem 13.265 · medium

Solve \( \displaystyle 2 \sin^{2}{\left(\theta \right)} - 3 \sin{\left(\theta \right)} + 1 = 0 \) for \( \displaystyle 0 \le \theta < 2\pi \).

\[ 2 \sin^{2}{\left(\theta \right)} - 3 \sin{\left(\theta \right)} + 1 = \left(\sin{\left(\theta \right)} - 1\right) \left(2 \sin{\left(\theta \right)} - 1\right) \]
Factor as a quadratic in sin θ.✓ Proved
Set each factor to zero; then find every angle in [0, 2π) with those values.
Reviewed
\[ \left. 2 \sin^{2}{\left(\theta \right)} - 3 \sin{\left(\theta \right)} + 1 \right|_{\substack{ \theta=\frac{\pi}{6} }} = 0 \]
θ = pi/6 works.✓ Proved
\[ \left. 2 \sin^{2}{\left(\theta \right)} - 3 \sin{\left(\theta \right)} + 1 \right|_{\substack{ \theta=\frac{\pi}{2} }} = 0 \]
θ = pi/2 works.✓ Proved
\[ \left. 2 \sin^{2}{\left(\theta \right)} - 3 \sin{\left(\theta \right)} + 1 \right|_{\substack{ \theta=\frac{5 \pi}{6} }} = 0 \]
θ = 5*pi/6 works.✓ Proved

Answer \( \theta = \frac{\pi}{6},\ \frac{\pi}{2},\ \frac{5 \pi}{6} \)

Lines: 4 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly factors the quadratic, identifies the valid sine values, and verifies the resulting angles within the specified domain.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly factors the quadratic, identifies the valid sine values, and verifies the resulting angles within the specified domain.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly factors the quadratic in sin(theta), identifies the two cases (sin(theta)=1 and sin(theta)=1/2), and lists all valid solutions within the specified interval [0, 2pi). The verification steps confirm the roots.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (misleading) 2026-10-15 — 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 crucial step of solving sin(θ) = 1 and sin(θ) = 1/2 to derive the values, leaving the student without the method to find the solutions.
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.