Traverse computation – recovering length from latitude: If the latitude of a traverse leg is known, the length of that leg can be obtained by multiplying the latitude by which trigonometric function of the reduced bearing?

Introduction / Context:
In traverse calculations, each line is resolved into latitude (northing/southing) and departure (easting/westing). Sometimes, partial information is available, such as the latitude along with the reduced bearing (RB), from which the full line length must be recovered.

Given Data / Assumptions:

• Latitude L_lat = length * cos(RB).
• Departure D_dep = length * sin(RB) (not directly used here).
• Reduced bearing is measured from the north or south towards east or west within 0° to 90°.

Concept / Approach:
From the fundamental relation L_lat = length * cos(RB), rearrange to length = L_lat / cos(RB) = L_lat * sec(RB). Thus, multiplying the latitude by the secant of the reduced bearing yields the traverse leg length. This is useful in missing data problems and in reconstruction of traverse elements from partial records.

Step-by-Step Solution:

Verification / Alternative check:
Check with an example: if RB = 30° and latitude = 86.6 m, then length = 86.6 * sec 30° = 86.6 * 1.1547 ≈ 100 m, consistent with a 100 m line having cos 30° = 0.866.

Why Other Options Are Wrong:

• Sine or tangent are associated with departure or departure/latitude ratio, not with recovering length from latitude alone.
• Cosine would give the projection, not the recovery of length.

Common Pitfalls:
Mixing whole-circle and reduced bearings; using the wrong quadrant sign; forgetting that secant grows large for RB near 90°, amplifying any error in latitude.

Final Answer:
Secant of the reduced bearing