13. Given a = 222, b = 318, c= 406; find A = 32° 57' 8". 14. Given a = 275.35, b=189.28, c=301.47; find A, B, C. Ans. A = 63° 30' 57"; B = 37° 58′ 20′′; C = 78° 30′ 43′′. 15. Given a = 5238, b= 5662, c = 9384; find A and B. Ans. A = 29° 17' 16"; B = 31° 55′ 31′′. 16. Given a = 317, b = 533, c=510; find A, B, C. Ans. A 35° 18' 0"; B = 76° 18′ 52"; C = 68° 23' 8". 1. Given a = 116.082, b = 100, C118° 15' 41". Ans. 5112.25. = 2. Given a 8, b=5, C60°. 17.3205. 3. Given b= 21.5, c = 30.456, A = 41° 22'. 216.372. 4. Given a = 72.3, A = 52° 35', B = 63° 17'. 2644.94. 5. Given b=100, A = 76° 38′ 13′′, C = 40° 5'. 3506.815. 12. Given a = 63.89, b = 138.24, c=121.15. 3869.2. MEASUREMENT OF HEIGHTS AND DISTANCES. 120. Definitions. One of the most important applications of Trigonometry is the determination of the heights and distances of objects which cannot be actually measured. The actual measurement, with scientific accuracy, of a line of any considerable length, is a very long and difficult operation. But the accurate measurement of an angle, with proper instruments, can be made with comparative ease and rapidity. By the aid of the Solution of Triangles we can determine : (1) The distance between points which are inaccessible. (2) The magnitude of angles which cannot be practically observed. (3) The relative heights of distant and inaccessible points. A vertical line is the line assumed by a plummet when freely suspended by a cord, and allowed to come to rest. A vertical plane is any plane containing a vertical line. A horizontal plane is a plane perpendicular to a vertical line. A vertical angle is one lying in a vertical plane. A horizontal angle is one lying in a horizontal plane. An angle of elevation is a vertical angle having one side horizontal and the other ascending. An angle of depression is a vertical angle having one side horizontal and the other descending. By distance is meant the horizontal distance, unless otherwise named. By height is meant the vertical height above or below the horizontal plane of the observer. For a description of the requisite instruments, and the method of using them, the student is referred to books on practical surveying.* * See Johnson's Surveying, Gillespie's Surveying, Clarke's Geodesy, Gore's Geodesy, etc. 121. To find the Height of an Object standing on a Horizontal Plane, the Base of the Object being Accessible. Let BC be a vertical object, such as a church spire or a tower. From the base C measure a horizon- A tal line CA. At the point A measure the angle of elevation CAB. 3. AC, the breadth of a river, is 100 feet. At the point A, on one bank, the angle of elevation of B, the top of a tree on the other bank directly opposite, is 25° 37'; find the height of the tree. Ans. 47.9 feet. 122. To find the Height and Distance of an Inaccessible Object on a Horizontal Plane. Let CD be the object, whose base D is inaccessible; and let it be required. to find the height CD, and its horizon tal distance from A, the nearest acces- B a A sible point. (1) At A in the horizontal line BAD observe the DAC = a; measure AB = a, and at B observe the ≤ DBC = ß. Then CA = = a sin B sin (a-B) (Art. 95) (2) When the line BA cannot be measured directly toward 1. A river 300 feet wide runs at the foot of a tower, which subtends an angle of 22° 30' at the edge of the remote bank; find the height of the tower. Ans. 124.26 feet. 2. At 360 feet from the foot of a steeple the elevation is half what it is at 135 feet; find its height. Ans. 180 feet. 3. A person standing on the bank of a river observes the angle subtended by a tree on the opposite bank to be 60°, and when he retires 40 feet from the river's bank he finds the angle to be 30°; find the height of the tree and the breadth of the river. Ans. 2013; 20. 4. What is the height of a hill whose angle of elevation, taken at the bottom, was 46°, and 100 yards farther off, on a level with the bottom, the angle was 31° ? Ans. 143.14 yards. 123. To find the Height of an Inaccessible Object situated above a Horizontal Plane, and its Height above the Plane. Let CD be the object, and let A and B be two points in the horizontal plane, and in the same vertical plane with CD. At A, in the horizontal line B BAE, observe the CAE = α, α A E and DAE = y; measure AB =a, and at B observe the 1. A man 6 feet high stands at a distance of 4 feet 9 inches from a lamp-post, and it is observed that his shadow is 19 feet long: find the height of the lamp. Ans. 7 feet. 2. A flagstaff, 25 feet high, stands on the top of a cliff, and from a point on the seashore the angles of elevation of the highest and lowest points of the flagstaff are observed to be 47° 12′ and 45° 13' respectively: find the height of the cliff. Ans. 348 feet. 3. A castle standing on the top of a cliff is observed from two stations at sea, which are in line with it; their |