Length of a parallel between two meridians at latitude λ For a sphere of radius R, the length of a parallel (small circle) at latitude λ between two meridians differing by Δlongitude (in radians) equals R * Δlongitude multiplied by which trigonometric factor?

Introduction / Context:
Map distances along parallels depend on latitude because parallels are smaller circles than the equator. Surveyors and navigators routinely apply the cosine factor to convert longitude differences into arc lengths at a given latitude.

Given Data / Assumptions:

• Spherical Earth of radius R (for simplicity).
• Longitude difference expressed in radians, Δλ.
• Latitude of the parallel is λ.

Concept / Approach:
The radius of the parallel at latitude λ is R * cos λ. Therefore, an angular separation Δλ (radians) along that parallel corresponds to an arc length of (R * cos λ) * Δλ. This introduces the cosine-latitude factor into many geodetic and cartographic formulas.

Step-by-Step Solution:
Compute effective parallel radius: r_parallel = R * cos λ.Arc length along the parallel: s = r_parallel * Δλ.Thus, s = R * Δλ * cos λ → the required factor is cos λ.

Verification / Alternative check:
At the equator (λ = 0°), cos λ = 1, so the parallel is the equator with full radius R, as expected. Near the poles, cos λ → 0, so east–west distances per degree of longitude shrink toward zero—consistent with convergence of meridians.

Why Other Options Are Wrong:

• sin λ, tan λ, cot λ, sec λ do not describe the radius of the parallel; only cos λ scales the equatorial radius to the latitude circle.

Common Pitfalls:
Using degrees directly without converting Δlongitude to radians; always convert to radians when computing arc length with s = R * θ.

Final Answer:
cos λ