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THE POPULAR EDUCATOR. 



Pig. 18. 



PLANE TRIGONOMETRY. VI. 



APPLICATION OF- TEIGONOMETEY TO MEASUKEMENT. 

 THE object of this lesson is rather to suggest than enumerate 

 the practical uses of the science. Apart from its connection with 

 Navigation upon which more will be said in the papers shortly 

 to be devoted to that subject Trigonometry is mainly employed 

 in the practical work of measuring (1) heights and distances, 

 (2) areas, and (3) contents of solids. By way of example we 

 will take one or two of its simpler applications to the measure- 

 ment of heights and distances, space forbidding even the 

 enumeration of the many problems which may arise in measur- 

 ing and surveying most 

 of which may, however, 

 be solved, directly or in- 

 directly, by the formulas 

 already arrived at. 



PROBLEM I. To find 

 the height of an accessible 

 object situated on a hori- 

 zontal plane (Fig. 18). 



Let the tower, E c, be 

 the object. Measure from 

 it a convenient distance, 

 ED, and observe the angle BAC. The right-angled triangle 

 ABO can now be calculated (B c = A B . tan. BAG; see Section 

 X.), one side, A B (equal to E D), and one angle being known. To 

 B c add E B, the height of the observer's eye above the horizontal 

 plane, and we obtain the height of the tower. 



EXERCISE 8. 



1. A person, whose eye is 5 ft. 6 in. above the ground, having receded 

 125 ft. from the base of a tower, finds that its angular elevation is 

 52 34'. Calculate its height. 



2. From the other side of a street 42 ft. wide, I observe that the 

 elevation of the front of a house is 49 28'. What is the height of 

 the house, the height of my eye being 5 ft. ? 



PROBLEM II. To find the distance on a horizontal plane of 

 an object of known height. 



Let the tower in Pig. 18 be the object, and its distance from 

 D i.e., the length of A B be the information sought. The 

 angle BAC being found as before, this case differs from the 

 preceding only in that a different side of the triangle is given, 

 and it is calculated with equal ease by the means pointed out 

 in Section X. 



EXERCISE 9. 



1. The angle which a man's height subtends nt the eye is 10'. If 

 his height is assumed to be 6 ft., calculate his distance. 



2. The pyramid of Cheops is 490 ft. high. From a distant point of 

 the plain on which it stands, the elevation of its apex is observed to 

 be 13 49'. Calculate its horizontal distance. 



PROBLEM III. To 

 find the height and 

 distance of an inacces- 

 sible object on a hori- 

 zontal plane (Fig. 19). 



The simplest way to 

 do this is to observe its 

 elevation at two points, 

 A and B, in line with 

 the object, measuring 

 their distance apart. 

 Let the observed angles 



Fig. 19. 



be a and respectively. 

 rule of sines, 



Angle A c B = j8 o ; whence, by the 



BC = ABX 



sin. (0 - o) ' 

 but F c = B c . sin. ; 



, 

 and similarly, F B = A B X 



sin. (j8 - o) 



sin. o cos. $ 

 - v 

 sm. (j8 - a) 



F c, added to height of observer's eye, gives the height, and 

 P B gives the distance of the object. 



EXERCISE 10. 



1. Wishing to ascertain the height of a church steeple, to which 

 close access cannot be had, I select two stations in line with it, 52 yds. 



apart. At those stations I find the elevations to be 58 14' and 36 42* 

 respectively. The height of my eye above the ground is 4ft. 6 in. 

 What is the height of the steeple ? 



2. What is the height of a hill, its angle of elevation at the bottom 

 being 52, while 300 yds. from the bottom, measured horizontally, its 

 elevation is found to be 28 30 7 ? 



Fig. 20. 



If the nature of the ground prevents 

 two observations being taken in line 

 with the object, they may be taken as 

 at A B in Fig. 20. Measure A B and the 

 angles BAC, ABC, and F A c, which we 

 will call a, 0, and ft respectively. (The 

 two former must be measured by a 

 sextant). Thea, since A c B = supplement of a -f j8, .'. sin. A C B 

 = sin. (a + 0) ; 



AC = 



' sin. (a + ) ' 

 But F c = A c . sin. ft ; 



. sin. |3 sin. ft 



sin. (a + ) 

 and similarly for the distance. 



PROBLEM IV. To find the distance of an inaccessible 

 object without measuring its 

 elevation, and whether on a 

 horizontal plane or not (Fig. 

 21). 



Let c be the object and A 

 the point of observation ; 

 select any other point, B, and 

 measure A B and the angles 

 CAB and ABC. AC may be 

 calculated by Section XXL, 

 par. 3. If B be taken so that 



Fig. 21. 



A B c be a right angle, the case, 

 of course, is still simpler. 



, 



EXERCISE 11. 



1. In order to ascertain the distance of a windmill on the opposite 

 side of a river, I observe the angle between the windmill aud a flag- 

 staff, which is 356 yds. distant, and find it to be 53 4'. Proceeding 

 to the flagstaff, I find the angle between the windmill and the first 

 station to be 49 10'. What is the distance of the windmill ? 



2. Wanting to know the breadth of a river, I measure along the 

 bank a base of 250 ft., the extremities of which we will call A and B. 

 At the extremity A I find the angle made by B and a tree on the oppo- 

 site bank is 63 31'; at the extremity B I find the angle between A and 

 the tree is 57 28'. What is the breadth of the river ? 



3. To find the distance of a battery from an outpost, I make 200 

 paces in a direction at right angles to the line which connects it with 

 the outpost, and then find that it makes an angle of 67 23' with the 

 outpost. How many paces is it distant from the latter ? 



PROBLEM V. To find the 

 distance from each other of two 

 inaccessible objects (Fig. 22). 



Let c and D be the objects. 

 Measure a base line, A B, and 

 observe the angles ABC, A B D, 



BAC, BAD. Calculate A c and 

 AD as in the last problem. 

 Then, since CAD = BAG 



BAD, we have the necessary 

 materials for calculating the 

 triangle A c D (Sec. XXL, par. 

 distance, c D. 



2). 



Fig. 22. 

 Hence we obtain the 



EXERCISE 12. . 



1. To ascertain the distance between two batteries in an enemy's 

 works, a base line of 500 yds. is measured, and the angles which each 

 battery makes with the base-line are observed to be 118 20' and 46 14' 

 at one extremity, and 88 48' and 33 12' at the other. What is their 

 distance apart ? 



There is an ingenious way of finding the converse of this 

 problem viz., the distance between A and B by observations 

 upon c and D, the distance between the latter being known. 

 Assume A B = 1000 ; then, on that supposition, calculate c D, 

 without reference to its real value. Then, as the calculated 

 value of c D is to the real value, so is 1000 to the real value 

 of AB, 



