Interpretation of given information:
The location of the theodolite (surveyor) at point A.
First stake (Point B)
Second stake (Point C).
Points B and C are on opposite sides of the river, and distance BC represents the width of the river.
For point B:
distance = 562
bearing = 16° (from north clockwise)
For point C:
departure (Easting) = 330.87'
Latitude (northing) 198.8'
equivalent to:
distance = √(330.87²+198.8²)
=386.0005
bearing=90-atan(198.8/330.87)=61.6135°
Let's form triangle ABC,
AB=562'
AC=386.0005'
∠BAC=61.6135-16=45.6135°
Use cosine rule to find distance BC
BC²=562²+386.0005²-2(562)(386.0005)cos(45.6135°)
=161354.304
BC=√(161354.304)=401.7'
Width of bridge = 401.7'
(Do check the calculations for accuracy because I used the Google calculator with which I am not familiar.)