Calculating Latitude and Departure from Reduced Bearing

In surveying, latitude and departure are fundamental concepts used to determine the change in north-south and east-west position of a line.

• Latitude: The orthographic projection of a survey line onto the North-South meridian. It represents the change in the North-South direction.
• Departure: The orthographic projection of a survey line onto the East-West line (perpendicular to the meridian). It represents the change in the East-West direction.

The signs of latitude and departure depend on the quadrant in which the line lies, as indicated by its bearing:

• North-East (NE): Latitude (+) , Departure (+)
• South-East (SE): Latitude (-) , Departure (+)
• South-West (SW): Latitude (-) , Departure (-)
• North-West (NW): Latitude (+) , Departure (-)

Given Information for Latitude and Departure Calculation

We are given the following information for line AB:

• Reduced Bearing of line AB: N60° W
• Length of line AB: 100 m

The reduced bearing N60° W tells us that the line lies in the North-West quadrant. The angle is 60 degrees measured from the North direction towards the West.

Formulas for Latitude and Departure

The latitude and departure of a line can be calculated using the following trigonometric formulas:

Latitude (\(L_{AB}\)) \( = \) Length of line AB (\(l_{AB}\)) \( \times \) \( \cos(\text{Reduced Bearing Angle}) \)

Departure (\(D_{AB}\)) \( = \) Length of line AB (\(l_{AB}\)) \( \times \) \( \sin(\text{Reduced Bearing Angle}) \)

Step-by-Step Calculation of Latitude and Departure

Using the given values:

Length \(l_{AB}\) \( = \) 100 m

Reduced Bearing Angle \( = \) 60°

Calculating Latitude:

\(L_{AB}\) \( = \) \(100 \times \cos(60^\circ)\)

We know that \( \cos(60^\circ) \) \( = \) 0.5

\(L_{AB}\) \( = \) \(100 \times 0.5\)

\(L_{AB}\) \( = \) 50 m

Since the bearing is N60° W (North-West quadrant), the Latitude is positive (+ve).

Latitude \( = \) \(+50\) m

Calculating Departure:

\(D_{AB}\) \( = \) \(100 \times \sin(60^\circ)\)

We know that \( \sin(60^\circ) \) \( = \) \( \frac{\sqrt{3}}{2} \) \( \approx \) 0.866

\(D_{AB}\) \( = \) \(100 \times 0.866\)

\(D_{AB}\) \( \approx \) 86.6 m

Since the bearing is N60° W (North-West quadrant), the Departure is negative (-ve).

Departure \( \approx \) \(-86.6\) m

Final Latitude and Departure Values

Based on the calculations, the latitude and departure of line AB are approximately +50 m and -86.6 m, respectively.

Latitude: +50 m

Departure: -86.6 m

Revision Table: Surveying Terms

Additional Information: Bearings in Surveying

Understanding different types of bearings is crucial in surveying:

• Whole Circle Bearing (WCB): Angle measured clockwise from the North direction. Values range from 0° to 360°. This system is commonly used as it avoids sign conventions for latitude and departure based on quadrant; trigonometric functions handle the signs automatically.
• Quadrantal Bearing (QB) or Reduced Bearing (RB): Angle measured from the nearest meridian (North or South) towards the East or West. Values range from 0° to 90°. It is expressed in the format like NθE, SθE, SθW, NθW, where θ is the acute angle.

To convert Whole Circle Bearing (WCB) to Reduced Bearing (RB):

• If WCB is in the 1st quadrant (0° to 90°), RB = WCB (N WCB E).
• If WCB is in the 2nd quadrant (90° to 180°), RB = 180° - WCB (S RB E).
• If WCB is in the 3rd quadrant (180° to 270°), RB = WCB - 180° (S RB W).
• If WCB is in the 4th quadrant (270° to 360°), RB = 360° - WCB (N RB W).

In this problem, the bearing N60° W is already given as a Reduced Bearing in the North-West quadrant.