SME
Surveying
Traverse computations
Use this for “A course runs 156.82 m at an azimuth of 127°40′00″. Find its latitude, departure and bearing.”
Diagram
lat = L cos(Az) = 157 × cos 128° = −95.8 m; dep = L sin(Az) = 124 m.
lat² + dep² = L²: north and east are +, south and west −.
This is an example. Type your own values to start a new problem.
lat (m)
=
L (m)
×
cos(
Az
)
dep (m)
=
L (m)
×
sin(
Az
)
β
Bearing
worked out (not typed)
Formulas
lat
=
L
×
cos(Az)
−95.8275 = 156.82 × cos(127°40′00″)
dep
=
L
×
sin(Az)
124.1354 = 156.82 × sin(127°40′00″)
L
=
√
lat² + dep²
156.82
=
√(
(−95.8275)
2
+
124.1354
2
)
β
=
the
bearing
of
Az
S 52°20′ E = the bearing of 127°40′00″
Az = atan2(dep, lat)
127°40′00″ = atan2(124.1354, −95.8275)
Assumptions
•
The azimuth is measured clockwise from north, 0° to 360°.
•
A bearing is the angle under 90° from north or south toward east or west.
•
North and east are +, south and west −.
Step-by-step
WE KNOW
Az = 127°40′00″, L = 156.82 m
FIND
latitude (lat), departure (dep), bearing (β)
Step 1 · Find latitude (lat)
Use lat = L × cos(Az)
The north part of the course: the side next to the azimuth.
lat = 156.82 × cos(127°40′00″)
lat = −95.8275 m
Step 2 · Find departure (dep)
Use dep = L × sin(Az)
The east part of the course: the side across from the azimuth.
dep = 156.82 × sin(127°40′00″)
dep = 124.1354 m
Step 3 · Find bearing (β)
Use β = the bearing of Az
A bearing is the angle under 90° from north or south, toward east or west: read it from the quarter the azimuth is in.
β
=
the
bearing
of
127°40′00″
127°40′00″ points south-east
180° − 127°40′00″ = 52°20′00″
127°40′00″ = S 52°20′ E
β
=
S
52°20′
E
Check
Put the numbers back in. Both sides should match.
−95.8275 = 156.82 × cos(127°40′00″) ✓
124.1354 = 156.82 × sin(127°40′00″) ✓
156.82
=
√(
(−95.8275)
2
+
124.1354
2
)
✓
127°40′00″ = S 52°20′ E ✓
127°40′00″
=
tan⁻¹(
124.1354
−95.8275
)
+
180°
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
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Use this for “A traverse 1250 m round sums to Σlat = −0.045 m and Σdep = +0.060 m. Find e, its direction and the precision.”
Compass-rule correction of one course
Use this for “Σlat = −0.045 m and Σdep = +0.060 m round 1250 m. Correct the 156.82 m course (lat −95.8275, dep 124.1354) by the compass rule.”
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