10. Computation of the sides of plane triangles (fig. 26). In the triangle AB a let the angles and the sides A B be given to find the other sides; i.e. let A B= 400, a=44°, B=96°, and A=40°. Then by the formula Supplying the radius and Ba= A B sin A sin a (p. 80). 11. To compute the angles and side of a plane triangle when two sides and a contained angle are given, let ABC be the angles of a plane triangle, and a b c the opposite sides. When the sides b c and the angle A are given, we have from trigonometry, Then Let b=400, c=370.13, and A=96°. (B+C)(=180—96) = 84°, (b+c)=770·13, (b−c)=29.87. Substituting these values in the formula, and taking 12. To compute for a side and two angles in a plane In the formula, if a be greater than 6 when B is the given angle, will be greater than 1, and therefore A b will be greater than B. It will be seen by an inspection of the diagram that A and 180-A, the exterior angle, are both greater than B, an interior angle of the triangle ABC. In this case two triangles, A B C, A' B C', fulfil the conditions, as the sin of an angle is equal the sin of its supple ment. Let a=860, b=640, and B=40°. Taking logarithms 13. To compute the angles of a plane triangle when the sides are given. The formula for finding the angle by the tan is c=980. = a+b+c sx (s-a) =s; let a=860, b=640, and From these we find s, and its excess above The angle may be also computed by the formula for sines, viz. Sin A a sin B (p 236). 14. For combined measured bases (fig. 27), the formula is, C C=AB√3" (p. 82). In a particular case let HK-C C'3. Then HK(=AB√33) =A B√27. Let A B 200. Then, taking logarithms = 15. Railway or circular curves :— a. To find the length of the tangent (fig. 36). Let the angle made by the tangents (B)=150° 0' 0" and R=80. Then, supplying the radius, the formula for the tangent is: R cot B r Bt (p. 110). Taking logarithms log R = 1.9030900 log cot B = 9.4280525 log Bt +10=11·3311425 log B te = 1·3311425 (= 21.436), the length of the tangent. |