Right spherical triangle (C = 90°) – choose the correct identity In a spherical triangle ABC right-angled at C, which identity is correct for sin b?

Introduction / Context:Right spherical triangles are central to many astro-survey calculations (e.g., azimuth–hour angle–latitude relationships). Napier’s rules and specialized identities simplify general spherical trigonometry when one angle (here C) is 90°.

Given Data / Assumptions:

• Triangle ABC with C = 90°.
• Sides a, b, c are opposite angles A, B, C respectively.

Concept / Approach:Using right spherical triangle relations, the sine rule reduces to convenient forms: sin a = sin A * sin c and sin b = sin B * sin c when C = 90°. These follow from general relationships with sin C = 1.

Step-by-Step Solution:Start from the spherical sine rule: sin a / sin A = sin b / sin B = sin c / sin C.With C = 90°, sin C = 1 → sin c / 1 = sin c.Therefore sin b = sin B * sin c (and similarly sin a = sin A * sin c).

Verification / Alternative check:Cross-check with Napier’s rules for right triangles; the same identity emerges consistently.

Why Other Options Are Wrong:They mix cosines or tangents in forms that do not correspond to the sine rule simplification for C = 90°.

Common Pitfalls:Swapping side–angle correspondences or applying plane trigonometric identities in place of spherical ones.

Final Answer:sin b = sin c * sin B