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Solving trigonometric equations

Problem 13.325 · easy

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

Isolate sin θ and find the angles in [0, 2π) with that value.
\[ \left. \sin{\left(\theta \right)} \right|_{\substack{ \theta=0 }} = 0 \]
θ = 0 works.✓ Proved
\[ \left. \sin{\left(\theta \right)} \right|_{\substack{ \theta=\pi }} = 0 \]
θ = pi works.✓ Proved

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

Lines: 2 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ 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 fails to identify all solutions in the given interval; theta = 0 and theta = pi are solutions, but theta = 2pi is excluded by the strict inequality, so the set is complete? Wait, sin(theta)=0 at 0, pi, 2pi, 3pi... In [0, 2pi), the solutions are 0 and pi. The stated answer is correct. However, the solution method is flawed because it just checks two values without explaining why these are the *only* solutions. It implies a guess-and-check method rather than a systematic derivation (e.g., using the unit circle or general solution formula). A student might learn that checking random points is sufficient, which is false for general problems. But strictly speaking, for this specific problem, the answer is correct. Is it an error? The prompt asks to review 'worked calculus solutions'. The 'sentence' says 'find the angles...'. The equations show two work. It does not prove uniqueness. This is a 'misleading' or 'error' in reasoning. Let's look closer. sin(theta)=0. General solution theta = n*pi. For 0 <= theta < 2pi, n=0 -> 0, n=1 -> pi, n=2 -> 2pi (excluded). So the answer is correct. The solution provided is incomplete reasoning. It doesn't show *how* they found them or why there aren't others. This is a 'misleading' pedagogical approach, or an 'error' in completeness. Given the adversarial role, I will flag it as an error because it lacks justification for completeness.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to identify all solutions in the given interval; theta = 0 and theta = pi are solutions, but theta = 2pi is excluded by the strict inequality, so the set is complete? Wait, sin(theta)=0 at 0, pi, 2pi, 3pi... In [0, 2pi), the solutions are 0 and pi. The stated answer is correct. However, the solution method is flawed because it just checks two values without explaining why these are the *only* solutions. It implies a guess-and-check method rather than a systematic derivation (e.g., using the unit circle or general solution formula). A student might learn that checking random points is sufficient, which is false for general problems. But strictly speaking, for this specific problem, the answer is correct. Is it an error? The prompt asks to review 'worked calculus solutions'. The 'sentence' says 'find the angles...'. The equations show two work. It does not prove uniqueness. This is a 'misleading' or 'error' in reasoning. Let's look closer. sin(theta)=0. General solution theta = n*pi. For 0 <= theta < 2pi, n=0 -> 0, n=1 -> pi, n=2 -> 2pi (excluded). So the answer is correct. The solution provided is incomplete reasoning. It doesn't show *how* they found them or why there aren't others. This is a 'misleading' pedagogical approach, or an 'error' in completeness. Given the adversarial role, I will flag it as an error because it lacks justification for completeness.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to identify all solutions in the given interval. It lists 0 and pi, but omits 2pi (which is excluded by strict inequality) and other potential roots? No, sin(theta)=0 at 0, pi, 2pi... wait. In [0, 2pi), the solutions are 0 and pi. Are there any others? No. So the answer is actually correct. Let me re-read carefully. Interval is 0 <= theta < 2pi. sin(0)=0, sin(pi)=0. sin(2pi)=0 but 2pi is not included. So the set {0, pi} is complete. Why did I think it was wrong? Ah, I need to check if the *method* is sound. The method says 'find the angles'. It checks two specific values. It does not explain *why* these are the *only* ones. It just verifies two candidates. This is not a valid derivation; it's a verification of a guess. A student would learn that checking two points is sufficient to solve an equation, which is false. The step 'find the angles' is not executed; it's just asserted via two checks. This is a logical gap/error in reasoning.
gpt-oss:20b: fail (error) 2026-10-16 — The solution only verifies that θ=0 and θ=π satisfy sinθ=0 but does not justify that these are the only solutions in [0,2π). It fails to establish the general solution sinθ=0 ⇒ θ=nπ and to rule out other possibilities. The setup is correct, but the reasoning is incomplete.
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution is incomplete because it fails to identify theta = 2pi as a solution, although 2pi is excluded by the strict inequality, the standard general solution or checking within the interval usually implies checking endpoints or understanding the periodicity. Wait, the interval is [0, 2pi). 2pi is not in the interval. So 0 and pi are the only solutions. However, the method described 'find the angles' is not executed systematically; it just guesses two values. A student might miss other solutions if they just guess. But more importantly, is there a missing solution? sin(theta)=0 at 0, pi, 2pi... In [0, 2pi), only 0 and pi are solutions. So the answer is correct. But the method is 'guess and check' without justification of completeness. Is that an error? It's misleading because it doesn't show how to find *all* solutions. It implies that checking 0 and pi is sufficient without explaining why no other angles work. This teaches bad habits.
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 2026-10-19 with SymPy 1.14.0.