Coordinates of O from S and D (FL/FR)

The horizontal angles are the means of face left and face right: at S, $\angle DSO = 75^\circ03'22''$ (both faces agree); at D, $\angle ODS = 68^\circ36'18''$ (both faces agree). O lies south of the line SD. Coordinates are (Easting, Northing).

        S ------------- D
         \            /
          \ 75°      / 68°
           \        /
            \      /
             \    /
              \  /
               O

Step 1: Length and bearing of the base SD

$$\Delta E = 2137.342,\quad \Delta N = -1565.199$$ $$SD = \sqrt{\Delta E^2 + \Delta N^2} = 2649.166\ \text{m}, \qquad \text{bearing of SD} = 126°12'57"\ (\text{south-east})$$

Step 2: Triangle SDO

$\angle SOD = 180^\circ - 75^\circ03'22'' - 68^\circ36'18'' = 36°20'20"$

By the sine rule:

$$SO = \frac{SD\sin 68^\circ36'18''}{\sin 36°20'20"} = 4162.625\ \text{m}, \qquad DO = \frac{SD\sin 75^\circ03'22''}{\sin 36°20'20"} = 4319.508\ \text{m}$$

Step 3: Bearings and coordinates

O is south of SD, i.e. on the right-hand side when facing from S to D, so the bearing of SO = bearing of SD + 75°03'22" $= 201°16'19"$.

$$E_O = E_S + SO\sin\theta = 627464.712 + 4162.625\sin 201°16'19" = 625954.540\ \text{m}$$ $$N_O = N_S + SO\cos\theta = 3066928.474 + 4162.625\cos 201°16'19" = 3063049.450\ \text{m}$$

Check from D: bearing of DO $= 306°12'57" - 68^\circ36'18'' = 237°36'39"$, giving $E = 625954.540$ m and $N = 3063049.450$ m (the same).

Answer: coordinates of O = (625954.540 E, 3063049.450 N).