Home›Calculus 1›Solving trigonometric equations›Problem 13.88

Solving trigonometric equations

Problem 13.88 · 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
\[ \left. 2 \sin^{2}{\left(\theta \right)} - 2 \sin{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }} = 0 \]
θ = pi/2 works.✓ Proved
\[ \left. 2 \sin^{2}{\left(\theta \right)} - 2 \sin{\left(\theta \right)} \right|_{\substack{ \theta=\pi }} = 0 \]
θ = pi works.✓ Proved

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

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 (style) — [domain objection, downgraded to style] 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 logical step of solving sin(θ) = 0 and sin(θ) = 1 for all possible angles in the given domain, leaving open the possibility of missing solutions (e.g., if the interval were larger) or including extraneous ones (though none are here, the method is incomplete).

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] 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 logical step of solving sin(θ) = 0 and sin(θ) = 1 for all possible angles in the given domain, leaving open the possibility of missing solutions (e.g., if the interval were larger) or including extraneous ones (though none are here, the method is incomplete).
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution verifies that the stated answers are roots, but it fails to demonstrate that these are the *only* solutions in the given interval. It omits the logical step of solving sin(theta) = 0 and sin(theta) = 1 for theta in [0, 2pi) to establish completeness.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution only verifies that the stated answers are solutions, but fails to demonstrate that they are the *only* solutions in the given interval. It omits the derivation of the general solutions from sin(θ) = 0 and sin(θ) = 1, leaving open the possibility of missing roots (e.g., if the interval were larger) or including extraneous ones.
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.