8.1 Latitudes, Departures, and Coordinates

8.1 Latitudes, Departures, and Coordinates

Quick Answer: Latitude of a course is ΔN = L cos(azimuth from north). Departure is ΔE = L sin(azimuth). North and east are positive. Convert a bearing to a north azimuth before you press COS/SIN: S 35°20' E is azimuth 144°40', not 35°20'. Adding latitudes and departures around a closed loop should return Σlat ≈ 0 and Σdep ≈ 0. The worked four-sided traverse ABCD is 1,770.20 ft around and misses by −0.19 ft north and +0.17 ft east.

Why this is on the CES exam

Domain III (Data Analysis and Evaluation) is 30% of the California Civil Engineering Surveying (CES) exam—the heaviest domain. Two professional activities in that domain are traverse calculations (closure, error, side shots) and rectangular coordinate system calculations. Knowledge item K is calculate rectangular coordinates (departures, latitudes); item B is error of closure. This section is the arithmetic those items rest on.

A California civil engineer uses these reductions to locate fixed works under Business and Professions Code (BPC) 6731.1—alignments, elevations, and the position of objects that support design and construction—not to run a Professional Land Surveyor (PLS) boundary traverse. The cosine/sine formulas are the same; the authority is not. Independent OpenExamPrep teaching here is for CES candidates working rectangular coordinates by hand or with a non-QWERTY calculator. Prometric allows up to two handheld calculators with no QWERTY keypad; the sign conventions still have to be in your head.

Assumed site coordinates from Chapter 5 use this same ΔN/ΔE arithmetic. Whether Hub A is N 5,000.00 on a warehouse-face meridian or a CCS83 northing, a course still breaks into latitude and departure in that system.

Latitude and departure

A course is one measured side of a traverse: a horizontal length L and a direction. Latitude is the north-south component. Departure is the east-west component.

Azimuth from north is measured clockwise from north: 0° = north, 90° = east, 180° = south, 270° = west. Check the identities on every exam sitting:

• Azimuth 0°: lat = +L, dep = 0
• Azimuth 90°: lat = 0, dep = +L
• Azimuth 180°: lat = −L, dep = 0
• Azimuth 270°: lat = 0, dep = −L

That is the entire NE + convention: north latitudes and east departures are positive. If you take cosine of a bearing angle without converting, S 35°20' E looks like a positive north latitude because cos(35°) is positive. It is not. The course goes south.

Bearing to azimuth

Back azimuth = forward azimuth ± 180° (bring the result into 0°–360°). You need it when you occupy the next station and want the reverse of the course you just measured.

Worked example: one course, bearing to coordinates

A taping crew reports course P–Q: bearing S 35°20' E, L = 250.00 ft. Station P is on an assumed site system at N = 1,000.00 ft, E = 2,000.00 ft.

1. Convert bearing to azimuth: 180° − 35°20' = 144°40' (144.6667°).
2. Latitude = 250.00 × cos(144.6667°) = 250.00 × (−0.815801) = −203.95 ft (south).
3. Departure = 250.00 × sin(144.6667°) = 250.00 × 0.578332 = +144.58 ft (east).
4. Coordinates of Q: N = 1,000.00 − 203.95 = 796.05 ft; E = 2,000.00 + 144.58 = 2,144.58 ft.

The same 250.00 ft on bearing N 35°20' E would be azimuth 35°20', latitude +203.95 ft, departure +144.58 ft. Sine and cosine magnitudes match; only the quadrant (the azimuth) supplies the signs. That is why CES items so often give a bearing and then ask for ΔN or ΔE—not because the trigonometry is exotic, but because the conversion is where people lose a sign.

Four-sided closed traverse ABCD

The rest of this chapter uses one loop so Sections 8.2 and 8.3 can reuse the same numbers. Interior angles were balanced as shown in Section 8.2 (a +4' raw angular misclosure, −1' on each angle). Distances are horizontal. Starting azimuth A→B is 42°18' from project north on an assumed datum. Station A is held at N = 5,000.00 ft, E = 2,000.00 ft.

Bearing-to-azimuth checks you should be able to write in the margin:

• AB: N 42°18' E → 42°18'
• BC: S 67°42' E → 180° − 67°42' = 112°18'
• CD: S 28°36' W → 180° + 28°36' = 208°36'
• DA: N 55°54' W → 360° − 55°54' = 304°06'

Latitudes and departures (hundredths of a foot)

Work each course at decimal degrees (42°18' = 42.30°, 67°42' = 67.70°, 28°36' = 28.60°, 55°54' = 55.90°), then round lat/dep to 0.01 ft.

AB at 42.30°: cos = 0.739631, sin = 0.673013. lat = 385.50 × 0.739631 = +285.13 ft dep = 385.50 × 0.673013 = +259.45 ft

BC at 112.30°: cos = −0.379456, sin = 0.925210. lat = 412.80 × (−0.379456) = −156.64 ft dep = 412.80 × 0.925210 = +381.93 ft

CD at 208.60°: cos = −0.877983, sin = −0.478692. lat = 468.20 × (−0.877983) = −411.07 ft dep = 468.20 × (−0.478692) = −224.12 ft

DA at 304.10°: cos = 0.560639, sin = −0.828060. lat = 503.70 × 0.560639 = +282.39 ft dep = 503.70 × (−0.828060) = −417.09 ft

Check the sums by grouping signs, not by re-keying four numbers:

• Σlat = (285.13 + 282.39) − (156.64 + 411.07) = 567.52 − 567.71 = −0.19 ft
• Σdep = (259.45 + 381.93) − (224.12 + 417.09) = 641.38 − 641.21 = +0.17 ft

A perfect loop would show both sums 0.00 ft. These residuals are the raw northing and easting error of closure. They look small next to 1,770 ft of measuring, which is why Section 8.2 converts them to a linear error and a precision ratio instead of stopping at −0.19 ft.

Unadjusted rectangular coordinates

Add each latitude to northing and each departure to easting, in order around the loop:

A′ is not A: ΔN = 4,999.81 − 5,000.00 = −0.19 ft, ΔE = 2,000.17 − 2,000.00 = +0.17 ft. That pair must match Σlat and Σdep. If they do not, you added the table wrong—fix the arithmetic before you talk about field error. Linear closure and Compass-rule adjustment in the next two sections start from this table, not from a new traverse.

Exam traps

• Pressing COS of the bearing 67°42' on course BC. Cos(67.70°) is positive; the course is south-east, so latitude must be negative. Convert to azimuth 112°18' first.
• Using slope distance as L. Latitude and departure use the horizontal length in the coordinate system you are working in (Chapter 5 and Chapter 9).
• Dropping a course from the Σlat / Σdep table, or adding a side shot into it (Section 8.3).
• Treating these independent teaching coordinates as a Board-published site. They are an assumed system for calculation practice.