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Solving trigonometric equations

Problem 13.89 · easy

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

Isolate cos θ and find the angles in [0, 2π) with that value.
\[ \left. \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }} = 0 \]
θ = pi/2 works.✓ Proved
\[ \left. \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{3 \pi}{2} }} = 0 \]
θ = 3*pi/2 works.✓ Proved

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

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 verifies that two values are solutions but fails to prove that they are the only solutions in the given interval. It does not use the periodicity or symmetry of the cosine function to establish uniqueness.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution verifies that two values are solutions but fails to prove that they are the only solutions in the given interval. It does not use the periodicity or symmetry of the cosine function to establish uniqueness.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution only verifies that the proposed answers are solutions, but fails to prove that they are the *only* solutions in the given interval. It does not demonstrate completeness.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to demonstrate that the listed solutions are the only ones in the interval [0, 2π). It merely verifies two specific values without establishing uniqueness or deriving them from the unit circle properties.
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.