Section 4. FORMULÆ FOR TRIGONOMETRICAL CAL CULATIONS. While the operations of the surveyor aim at obtaining a sufficient number of distances. angles, and heights, from which the required plan can be made, the whole of these are not necessarily measured, as some may be calculated from others. Again, calculation may also be adopted for checking the correctness of measurement. The greater part of the calculations of the surveyor are based on the principles of simple plane and spherical trigonometry, a few mensuration formulæ only being required for calculating areas of estates. A knowledge of these principles and their application forming part of ordinary education, the reader of this book will be assumed to possess this knowledge, while the formulæ more useful for the purposes of surveyor's triangulation will be found in the following collection. Occasional formulæ suited to particular instances will be given throughout the book attached to the case or special subject on which they bear. Collection of Trigonometrical Formulæ. I. Functions of any two angles. sin (A+B)=sin A cos B+cos A sin B sin (A-B)=sin A cos B-cos A sin B cos (A+B)=cos A cos B-sin A sin B cos (A-B)=cos A cos B+ sin A sin B tan (A+B)=(tan A+tan B)÷(1−tan A. tan B) tan (AB)=(tan A-tan B)÷(1+tan A. tan B) cotg A+B=(cotg A. cotg B–I)-(cotg A+cotg E) cotg A–B=(cotg A. cotg B+I)+(cotg B–cotg 4) tan Atan B tan A-tan B = sin (A+B)(cos A. cos B) sin (AB)-(cos A. cos B) =sin(A+B)(sin A. sin B) cotg B–cotg A =sin(A–B)(sin A. sin B) cotg A+ cotg B sin2 A-sin2 B cos2 A-sin2 B -sin A-B. sin A+B = =cos A-B cos A+B = tan2 A-tan B sin A-B. sin A+B+(cos2 A. cos2 B) cotgź B–cotg A=sin A– B.sin A+ B-(sin* A sin* B). II. The solution of triangles. Let a, b, c represent the three sides, and A, B, C, the three angles opposite to them of any plane triangle, 1 The algebraic signs of sines and cosecants of supplements are the same as those of the original angles; in the other quadrants the signs are given in the following table. a+b+c and let s= then the relations between the 2 functions of three angles are, sin Asin B+C; cos A=-cos B+C; Small angles are obtained through their sines, tangents or cosecants, and angles near 90° through the other functions. In the ambiguous solution, refer to the field record as to whether the angle was obtuse or acute. A right-angled triangle forms a particular case of the above. The following formula affords a check on the correctness of the observed inward angles of any irregular polygon of traverse. Let N the number of angles of the polygon. Then in the particular case of the angles being all salient, the sum of all the observed inward angles should be=2N × 90° - 360. And if there be any re-entering angles the sum of all the salient inward angles should be equal to the sum of all the re-entering outward angles. III. The calculation of areas. where d is the perpendicular from A on a. = Parallelogram; area bdbc. sin A, where d is the perpendicular on b. d Trapezoid; area=(a+b) a, where a and b are parallel at a distance apart=d. Regular polygon; area= cotg na2 180° n where = number of sides, each of which=a. b=their common distances apart. Irregular figure slightly curved (parabolic) b in this case n must be an even number. IV. The solution of geodetic or large triangles on the earth's surface. 1. For limited purposes of this description, up to triangles of 40 miles in length of side, the earth is generally considered an exact spheroid, having the ratio of 99666 for its polar and mean equatorial diameters, or an ellipticity of about, and a mean radius of 20 900 086 feet; on this assumption, 100000 Let R the mean radius of curvature of the earth in feet, P the area of the large triangle (calculated as plane) in square feet, E the spherical excess in seconds, P.648 000 Then E= or log E-log P-9'325871. TR2 2. To apportion the errors of observation among the angles. Having obtained the spherical excess, and using mean values of A, B, and C, then A+B+C should =180° + E; and any small error indicated should be divided equally among the three angles, if the number of observations of each have been equal. But if otherwise, Gauss's rule must be used to apportion the error (T) among the angles. Let the number of recorded observations of A, 1, l', ', &c. the seconds of reading of the several observations, m=the mean value or average of them, x=the actual amount of error in A, Then a= And (m −1)2 + (m—l')2 + (m − l'')2 + &c. n2 and y the corresponding quantities may in the same way be found for the angles B and C ; and their sum a+B+y obtained. Then if y and z are the actual errors in B and C If a single angle (C) has been only once observed, an arbitrary value is assumed for y. 3. For the computation of the sides of such a triangle, reduce each angle after correction by one-third of the spherical excess thus calculated, but keep any measured side unaltered in length; the remaining two sides may then be calculated by the formulæ given for plane triangles, without any error whatever. E |