Tacheometry with inclined sights (staff held normal): sign convention in the horizontal distance formula In the expression D = s cos θ + (f + d) cos θ ± h sin θ for horizontal distance with staff held normal to the line of sight, which sign applies to the term h sin θ for angle of elevation vs. depression?

Introduction / Context:
In tacheometric distance reduction with inclined sights and staff normal to the line of sight, a small correction term involving the instrument height h appears. Correct sign handling is necessary to compute horizontal distance D accurately for both upward and downward sights, especially on steep ground.

Given Data / Assumptions:

• θ is the vertical angle (positive for elevation, negative for depression by convention).
• s is the staff intercept; f is focal length; d is the distance from objective to instrument axis; h is the instrument height above ground at the instrument station.
• Staff is normal to the line of sight.

Concept / Approach:

When the line of sight is elevated, the horizontal projection from instrument axis to the staff foot is reduced by h sin θ; when depressed, it increases by h sin |θ|. Hence the sign is negative for elevation and positive for depression. Many practical texts present the compact ± notation with the upper (−) sign for elevation and lower (+) sign for depression.

Step-by-Step Solution:

Verification / Alternative check:

Check limiting case θ = 0: term vanishes. For symmetric ±θ slopes to same point at equal height difference, signs are opposite, matching field experience.

Why Other Options Are Wrong:

Options A and E contradict the standard convention. D is incorrect as a generic statement; B states the correct case only partially.

Common Pitfalls:

Mixing sign conventions from different textbooks; forgetting that h refers to instrument height, not staff height; applying the staff-vertical formula to the staff-normal case.

Final Answer:

minus sign is used for angle of elevation