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Solving trigonometric equations

Problem 13.409 · easy

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

Isolate cos θ and find the angles in [0, 2π) with that value.
\[ \left. 2 \cos{\left(\theta \right)} - \sqrt{2} \right|_{\substack{ \theta=\frac{\pi}{4} }} = 0 \]
θ = pi/4 works.✓ Proved
\[ \left. 2 \cos{\left(\theta \right)} - \sqrt{2} \right|_{\substack{ \theta=\frac{7 \pi}{4} }} = 0 \]
θ = 7*pi/4 works.✓ Proved

Answer \( \theta = \frac{\pi}{4},\ \frac{7 \pi}{4} \)

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 only verifies that the proposed answers are solutions, but it fails to demonstrate that these are the *only* solutions in the given interval. A complete solution requires deriving the general solution or explicitly ruling out other quadrants.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-21 — The solution only verifies that the proposed answers are solutions, but it fails to demonstrate that these are the *only* solutions in the given interval. A complete solution requires deriving the general solution or explicitly ruling out other quadrants.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution only verifies that the proposed answers are solutions, but fails to demonstrate that they are the *only* solutions in the given interval. It does not perform the necessary derivation (isolating cos(theta) and using the unit circle) to establish completeness.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly isolates cos(θ) = √2/2 and identifies the two solutions in the interval [0, 2π) where cosine is positive (Quadrants I and IV). The verification steps confirm the answers.
gpt-oss:20b: pass 2026-10-20

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 2026-10-21 with SymPy 1.14.0.