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Exact trigonometric values

Problem 13.413 · easy

Find the exact value of \( \displaystyle \sin{\left(\frac{7 \pi}{4} \right)} \).

The angle 7*pi/4 ends in quadrant 4 with reference angle pi/4, where sin is negative.
Reviewed
\[ \sin{\left(\frac{\pi}{4} \right)} = \frac{\sqrt{2}}{2} \]
sin of the reference angle.✓ Proved
\[ \sin{\left(\frac{7 \pi}{4} \right)} = - \frac{\sqrt{2}}{2} \]
Attach the sign for the quadrant.✓ Proved

Answer \( - \frac{\sqrt{2}}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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	mpmath evaluates the function at the angle

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the quadrant, reference angle, and sign convention to determine the exact value.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly identifies the quadrant, reference angle, and sign convention to determine the exact value.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly identifies the quadrant, reference angle, and sign convention. The logic is sound and the result is correct.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly identifies the quadrant, reference angle, and sign convention to derive the exact value.
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_values, checked 2026-10-21 with SymPy 1.14.0.