How many latitude and departure for line when it has bearing S50°E and length 50m ?

Question

Examoogle · Accepted Answer

The correct answer is Option C: $L = 32.13	ext{ (S)}$, $D = 38.30	ext{ (E)}$. The latitude ($L$) and departure ($D$) of a line are its orthogonal components along the North-South and East-West meridian directions, respectively. For a line of length $l$ and reduced bearing $	heta$ in the SE quadrant (S $	heta$ E), the latitude is directed South ($l \cos 	heta$) and the departure is directed East ($l \sin 	heta$).

Step-by-Step Solution & Concepts:
The latitude ($L$) and departure ($D$) of a line are its orthogonal components along the North-South and East-West meridian directions, respectively. For a line of length $l$ and reduced bearing $	heta$ in the SE quadrant (S $	heta$ E), the latitude is directed South ($l \cos 	heta$) and the departure is directed East ($l \sin 	heta$).

Working Principle: Latitude is the length of the projection of a line on the reference meridian (North-South line), calculated as $L = l \cos 	heta$. Departure is the length of the projection on the perpendicular axis (East-West line), calculated as $D = l \sin 	heta$. Since the bearing S50°E falls in the second quadrant (Southeast), the latitude is negative (South) and the departure is positive (East).

Key Formulas: $	ext{Latitude } (L) = l \cos 	heta$ — Projection of line on the North-South axis; $	ext{Departure } (D) = l \sin 	heta$ — Projection of line on the East-West axis

Key Points: Latitude is positive in North quadrant and negative in South quadrant.; Departure is positive in East quadrant and negative in West quadrant.; For bearing S50°E: Latitude is directed South (S) and Departure is directed East (E).; Calculation: $L = 50 	imes \cos(50^\circ) = 32.139	ext{ m (S)}$, $D = 50 	imes \sin(50^\circ) = 38.302	ext{ m (E)}$.

Applications: Computation of consecutive coordinates in traverse surveying; Checking traverse closure error and adjusting closed traverses

Additional Info: Option A gives North latitude and West departure, which corresponds to N50°W.. Option B gives North latitude and East departure, which corresponds to N50°E.. Option D gives South latitude and West departure, which corresponds to S50°W.

Answer: C is correct — For a line bearing S50°E with length 50m, the latitude is $50 \cos 50^\circ = 32.13	ext{ m (South)}$ and departure is $50 \sin 50^\circ = 38.30	ext{ m (East)}$.

Exam Tip: Always identify the quadrant of the Reduced Bearing (RB) first to fix the directional signs (N/S for Latitude, E/W for Departure) before computing the trigonometric values.

Examoogle · Answer

Option A: $L = 32.13	ext{ (N)}$, $D = 38.30	ext{ (W)}$

Examoogle · Answer

Option B: $L = 32.13	ext{ (N)}$, $D = 38.30	ext{ (E)}$

Examoogle · Answer

Option D: $L = 32.13	ext{ (S)}$, $D = 38.30	ext{ (W)}$

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