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Solving trigonometric equations

Problem 13.78 · medium

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

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

Answer \( \theta = \frac{\pi}{6},\ \frac{5 \pi}{6},\ \frac{3 \pi}{2} \)

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 that the resulting angles satisfy the original equation within the specified domain.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly factors the quadratic, identifies the valid sine values, and verifies that the resulting angles satisfy the original equation within the specified domain.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution verifies that the listed values are roots but fails to demonstrate that they are the *only* solutions in the interval [0, 2π). It omits the crucial step of solving sin(θ) = -1 and sin(θ) = 1/2 to derive the angles, leaving a logical gap between the factors and the specific answers.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly factors the quadratic in sin(theta), identifies the correct reference angles and quadrants for the given interval, and verifies the solutions.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.