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Solving trigonometric equations

Problem 13.480 · medium

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

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

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

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 fails to solve for theta; it only verifies that the stated answers are roots. It does not demonstrate that these are the *only* solutions in the interval, nor does it derive them from the factors sin(theta)=0 and sin(theta)=-1.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-09 — The solution fails to solve for theta; it only verifies that the stated answers are roots. It does not demonstrate that these are the *only* solutions in the interval, nor does it derive them from the factors sin(theta)=0 and sin(theta)=-1.
gpt-oss:20b: pass 2026-11-09
qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution verifies that the stated answers are roots but fails to demonstrate that they are the *only* solutions in the interval [0, 2π). It omits the derivation of the candidate values (sin(θ) = 0 and sin(θ) = -1) from the factored equation, leaving the completeness of the solution unjustified.
gpt-oss:20b: pass 2026-11-08
qwen3.6:27b-mlx: fail (error) 2026-11-08 — The solution verifies that the stated answers are roots but fails to demonstrate that they are the *only* solutions in the given interval. Specifically, it does not explicitly solve sin(θ) = -1 to derive θ = 3π/2, nor does it rule out other potential solutions, making the logic incomplete.
gpt-oss:20b: pass 2026-11-08

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 2026-11-09 with SymPy 1.14.0.