Sextant observation to two elevated objects: computing the true horizontal angle Two objects A and B have elevations (altitudes) α and β respectively. A sextant measures the angle θ between the lines of sight to A and B. What is the correct horizontal angle H between A and B at the observer?

Introduction / Context:
When two targets lie at elevations α and β, the sextant (or any angle-measuring device) reads the included angle θ between the two sloping sight lines. However, mapping and traverse work often requires the true horizontal angle H. Converting the observed sloping angle to its horizontal equivalent is a classic spherical/space-geometry problem that uses vector dot products or direction cosines.

Given Data / Assumptions:

• Elevations (altitudes) of A and B above the horizontal through the observer are α and β.
• Observed included angle between the two sight lines is θ.
• Atmospheric refraction and instrumental errors are neglected.

Concept / Approach:

Let unit vectors to the two objects be u and v. Using elevation (altitude) and azimuth notation, the cosine of the angle γ between u and v is given by:
cos γ = sin α * sin β + cos α * cos β * cos H, where H is the unknown horizontal angle between their azimuths. Since the sextant measures γ = θ, we solve for cos H. Rearranging gives the fundamental relation:
cos H = (cos θ − sin α * sin β) / (cos α * cos β).

Step-by-Step Solution:

Verification / Alternative check:

For α = β = 0 (both targets on the horizon), the formula reduces to cos H = cos θ → H = θ, as expected. For small elevations, a first-order expansion yields H ≈ θ − (tan(α/2) * tan(β/2)) * sin θ (small correction), consistent with practical experience.

Why Other Options Are Wrong:

Option B uses a plus sign; this would make H smaller or imaginary under typical geometries and fails the horizon check.

Option C is an unjustified linear scaling and fails dimensional tests; it does not arise from the vector identity.

Option D is incorrect because option A provides the correct closed-form expression.

Common Pitfalls:

Forgetting to convert degrees to radians in software; misinterpreting α and β as zenith angles (use altitude elevations here); ignoring quadrant logic when using arccos in coding environments.

Final Answer:

H = arccos{ (cos θ − sin α * sin β) / (cos α * cos β) }