Home›Calculus 1›Solving trigonometric equations›Problem 13.90

Solving trigonometric equations

Problem 13.90 · easy

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

Isolate sin θ and find the angles in [0, 2π) with that value.
Reviewed
\[ \left. 2 \sin{\left(\theta \right)} + \sqrt{2} \right|_{\substack{ \theta=\frac{5 \pi}{4} }} = 0 \]
θ = 5*pi/4 works.✓ Proved
\[ \left. 2 \sin{\left(\theta \right)} + \sqrt{2} \right|_{\substack{ \theta=\frac{7 \pi}{4} }} = 0 \]
θ = 7*pi/4 works.✓ Proved

Answer \( \theta = \frac{5 \pi}{4},\ \frac{7 \pi}{4} \)

Lines: 2 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the target value for sine and provides the correct angles within the specified interval. Although it verifies the answers rather than deriving them, the logic is sound and the result is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the target value for sine and provides the correct angles within the specified interval. Although it verifies the answers rather than deriving them, the logic is sound and the result is correct.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to actually solve for theta; it only verifies the given answers without showing the derivation (e.g., finding the reference angle and determining quadrants). This does not constitute a solution.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the reference angle and the quadrants where sine is negative, providing the correct solutions within the specified interval.
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.