. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. e base is 148 feet, andperpendicular 45 feet ? Ans. 370 yards. 2. What is the area of a triangle whose base is 18i feet and alti-tude 25\ feet ? Ans. 231^1 feet. PROBLEM III. Investigate and give a rule for finding the area of a triangle whentwo sides and t/ieir included angle are given. Let ABC be the triangles, AB, BO thegiven sides, and B the given angle. Represent the side opposite to the angle A,by a, opposite C, by c, and opposite B, by b. Now a and c are the
. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. e base is 148 feet, andperpendicular 45 feet ? Ans. 370 yards. 2. What is the area of a triangle whose base is 18i feet and alti-tude 25\ feet ? Ans. 231^1 feet. PROBLEM III. Investigate and give a rule for finding the area of a triangle whentwo sides and t/ieir included angle are given. Let ABC be the triangles, AB, BO thegiven sides, and B the given angle. Represent the side opposite to the angle A,by a, opposite C, by c, and opposite B, by b. Now a and c are the given sides, and by problem II, the area is ya{AD) (l) The trigometrical value of AD can be found from the rightangled triangle ABD. Thus, sin. ABB : c : : sin. B : AD. That is, 1 : c : : sin. B : AD. Whence AD=c sin. B. This value of AD substituted in (1) gives \ac sin. B= area A (2). This expression is the area of the triangle, and from it we drawthe following rule. Rule.— Take half the product of the two given sides and multiplyit by the natural sine of the included angle, and the last product will bethe area 72 SURVEYING. 1. One side of a triangle is 82 feet, another 90 feet, and theirincluded angle is 27° 31. What is the area ? Ans. square ° 31 Nat. sine. - - - 46201\ac - - - - 3780 3996080323407138603 Ans. When we use logarithms we have the following rule : Rule.— To the logarithms of the two sides, add the log. sine of theincluded angle, and the sum rejecting 10, in the index, is the logarithmof twice the area of the triangle. 2. A certain triangle has one side , another equal ,and their included angle 57° 25, what is its area ? Ans. log. log. 57° 25 sine 2 Area, log. 3. How many square yards in a triangle, two sides of which are25 and 21J- feet, and their included angle 45° ? Ans. PROBLEM IV. Investigate and give a rule for finding the area of
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