What are the latitude and the departure of a 300 m traverse line with a bearing of 240°?

🧭 Understanding Latitude and Departure

In traversing, the position of a survey line is defined by its length and direction. To perform calculations, this line is resolved into two perpendicular components:

• Latitude: The projection of the line onto the North-South (N-S) axis. It's how far North or South the line extends. North latitudes are positive (+), and South latitudes are negative (-).
• Departure: The projection of the line onto the East-West (E-W) axis. It's how far East or West the line extends. East departures are positive (+), and West departures are negative (-).

🔬 Step-by-Step Calculation

The calculation involves converting the bearing to the correct format and then applying trigonometric formulas.

Step 1: Convert Whole Circle Bearing (WCB) to Reduced Bearing (RB)

The formulas for latitude and departure use the angle from the N-S meridian, which is found in the Reduced Bearing (RB) system.

• Given Length (L) = 300 m
• Given Whole Circle Bearing (WCB) = 240°

A bearing of 240° is in the third quadrant (between 180° and 270°). The line is measured from the South meridian towards the West.

Reduced Bearing Angle (θ) = 240° - 180° = 60°

So, the Reduced Bearing is S 60° W.

Step 2: Calculate the Latitude

The formula for latitude is L cos(θ).

Latitude = 300 × cos(60°)

Since cos(60°) = 0.5:

Latitude = 300 × 0.5 = 150 m

Because the bearing is in the South direction, the latitude is negative.

Latitude = -150 m

Step 3: Calculate the Departure

The formula for departure is L sin(θ).

Departure = 300 × sin(60°)

Since sin(60°) = √3 / 2:

Departure = 300 × (√3 / 2) = 150√3 m

To match the format in the options (dividing by √3), we can write this as:

150√3 = (150 × 3) / √3 = 450/√3 m

Because the bearing is in the West direction, the departure is negative.

Departure = -450/√3 m