Slope correction in chaining: If the measured chain length along a slope of angle θ is l, what is the slope correction to reduce it to horizontal length?

Introduction / Context:
Distances for plans and coordinates are horizontal. When a line is measured on a slope, a correction must be applied to obtain the equivalent horizontal length. A convenient trigonometric identity expresses this correction in terms of half-angle functions, which is especially useful with small slopes.

Given Data / Assumptions:

• Slope angle = θ (between line and horizontal).
• Measured length along slope = l.
• Goal: compute correction to subtract from l to get horizontal projection.

Concept / Approach:
Horizontal length = l * cos θ. Therefore slope correction C_s = l − l * cos θ = l * (1 − cos θ). Using the identity 1 − cos θ = 2 * sin^2(θ/2), we get C_s = 2 * l * sin^2(θ/2). This quantity is subtracted from the measured sloping length to obtain the horizontal distance.

Step-by-Step Solution:

Verification / Alternative check:
For small θ, sin(θ/2) ≈ θ/2 (in radians), so C_s ≈ 2 * l * (θ^2 / 4) = l * θ^2 / 2, matching small-angle approximations used in quick estimates.

Why Other Options Are Wrong:

• cos^2(θ/2) forms (options a, d) do not equal 1 − cos θ.
• tan^2(θ/2) alone does not represent the needed difference without a denominator factor.

Common Pitfalls:
Applying the correction with wrong sign; the sloping distance is always longer than horizontal for θ ≠ 0, so subtract C_s.

Final Answer:
2 * l * sin^2(θ/2)