example, a line with an azimuth of 341, 12' 30 " falls in the fourth or northwest quadrant and its bearing
is N 18, 47' 30 "W.
s. Coordinate Computations. If the coordinate of a point and the azimuth and distance from that
point to a second point are known, surveyors can compute the coordinate of the second point. The
azimuth and distance from Station A to Station B are determined by measuring the horizontal angle from
the azimuth mark to Station B and the distance from Station A to Station B.
(1) A grid is a rectangular system in which the easting and northing lines form right angles at the
point of intersection. The computation of the difference in northing (dN)(side Y) and the difference in
easting (dE)(side X) uses the formulas for the computation of a right triangle. The distance from Station
A to Station B is the hypotenuse of the triangle, and the bearing angle (azimuth) is the known angle.
The following formulas are used to compute dN and dE:
dN = cosine azimuth x distance
dE = sine azimuth x distance
(2) If the traverse leg falls in the first (northeast) quadrant, the value of the easting increases as
the line goes east, and the value of the northing increases as it goes north. The product of the dE and the
dN are positive and are added to the easting and northing of Station A to obtain the coordinate of Station
(3) When surveyors use calculators with trigonometric functions to compute the traverse, the
azimuth is entered and the calculator provides the correct sign of the function, the dN, and the dE. If the
functions are taken from tables, the computer provides the sign of the function based on the quadrant.
Lines going north have positive dNs, and lines going south have negative dNs. Lines going east have
positive dEs, and lines going west have negative dEs. The formulas shown in the following examples
are used to determine the dN and the dE:
Example 1: Given an azimuth from Station A to Station B of 70, 15' 15" and a distance of 568.78
meters (this falls in the first [northeast] quadrant), compute the dN and the dE.
dN = cosine 70 15' 15" x 568.78 meters
= +0.337848 x 568.78 meters = +192.16 meters
dE = sine 70 15' 15" x 568.78 meters
= +0.941200 x 568.78 = +535.34 meters
Example 2: Given an azimuth from Station B to Station C of 161, 12' 30" and a distance of 548.74
meters (this falls in the second [southeast] quadrant), compute the dN and the dE.
dN = cosine 161 12' 30" x 548.74 meters
= -0.946696 x 548.74 meters = -519.49 meters