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Solving trigonometric equations

Problem 13.79 · medium

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

Let u = 2θ. Since 0 ≤ θ < 2π, u runs over [0, 4π): two full turns, so twice as many solutions.
\[ \left. \sin{\left(2 \theta \right)} - \frac{\sqrt{2}}{2} \right|_{\substack{ \theta=\frac{\pi}{8} }} = 0 \]
θ = pi/8 works.✓ Proved
\[ \left. \sin{\left(2 \theta \right)} - \frac{\sqrt{2}}{2} \right|_{\substack{ \theta=\frac{3 \pi}{8} }} = 0 \]
θ = 3*pi/8 works.✓ Proved
\[ \left. \sin{\left(2 \theta \right)} - \frac{\sqrt{2}}{2} \right|_{\substack{ \theta=\frac{9 \pi}{8} }} = 0 \]
θ = 9*pi/8 works.✓ Proved
\[ \left. \sin{\left(2 \theta \right)} - \frac{\sqrt{2}}{2} \right|_{\substack{ \theta=\frac{11 \pi}{8} }} = 0 \]
θ = 11*pi/8 works.✓ Proved

Answer \( \theta = \frac{\pi}{8},\ \frac{3 \pi}{8},\ \frac{9 \pi}{8},\ \frac{11 \pi}{8} \)

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	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
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 4 roots in [0, 2π), the same ones

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution only verifies that the four listed values are solutions but fails to prove that there are no other solutions. It does not derive the general solution or explicitly check all quadrants for u in [0, 4pi), leaving the completeness of the answer unjustified.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution only verifies that the four listed values are solutions but fails to prove that there are no other solutions. It does not derive the general solution or explicitly check all quadrants for u in [0, 4pi), leaving the completeness of the answer unjustified.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution only verifies that the stated answers are correct but fails to demonstrate that they are the *only* solutions. It does not solve the equation or argue for completeness within the given interval.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution merely verifies the four proposed answers without deriving them or proving they are the only solutions. It fails to show the logical steps (solving for the reference angle and determining all quadrants within the expanded domain) required to solve the equation.
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.