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Exact values of inverse trig functions

Problem 13.311 · easy

Find the exact value of \( \displaystyle \arcsin{\left(- \frac{\sqrt{3}}{2} \right)} \).

arcsin returns the one angle in [−π/2, π/2] whose sin is -sqrt(3)/2.
Reviewed
\[ \sin{\left(\frac{\left(-1\right) \pi}{3} \right)} = - \frac{\sqrt{3}}{2} \]
sin(-pi/3) = -sqrt(3)/2, and -pi/3 is in [−π/2, π/2].✓ Proved
\[ \operatorname{asin}{\left(\frac{\left(-1\right) \sqrt{3}}{2} \right)} = - \frac{\pi}{3} \]
So that angle is the value.✓ Proved

Answer \( - \frac{\pi}{3} \)

✓ 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 angle lies within this range and satisfies the sine equation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — 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 2026-10-19
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 sine equation.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15
gpt-oss:20b: pass 2026-10-15

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 2026-10-19 with SymPy 1.14.0.