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Exact values of inverse trig functions

Problem 13.180 · easy

Find the exact value of \( \displaystyle \arcsin{\left(- \frac{\sqrt{2}}{2} \right)} \).

arcsin returns the one angle in [−π/2, π/2] whose sin is -sqrt(2)/2.
Reviewed
\[ \sin{\left(\frac{\left(-1\right) \pi}{4} \right)} = - \frac{\sqrt{2}}{2} \]
sin(-pi/4) = -sqrt(2)/2, and -pi/4 is in [−π/2, π/2].✓ Proved
\[ \operatorname{asin}{\left(\frac{\left(-1\right) \sqrt{2}}{2} \right)} = - \frac{\pi}{4} \]
So that angle is the value.✓ Proved

Answer \( - \frac{\pi}{4} \)

✓ 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's inverse function

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the range of the arcsine function and verifies that the proposed answer lies within this range and satisfies the equation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the range of the arcsine function and verifies that the proposed answer lies within this range and satisfies the equation.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the range of the arcsine function and verifies that the candidate angle lies within this range and satisfies the equation.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the range of the arcsine function and verifies that the proposed angle lies within this range and satisfies the sine 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/inverse_trig_values, checked 2026-10-18 with SymPy 1.14.0.