D Thus, if B be the point of observation (Fig. 68) and A the A given object, the angle made by the line joining B to A, with the horizontal line BC, is called the angle of elevation. In a similar manner if A be the point of observation and B an object, then the angle between the horizontal line (DA, drawn through A) and the line AB is called the angle of depression. In all cases the diagrams made should be as clear and accurate as possible. Even when the result depends alone on calculation it can, and should be, roughly checked by graphic construction. The following problems will show the methods adopted. Ex. 1. At a distance of 100 feet from the foot of a tower the angle of elevation of top of tower is found to be 60°. Find the height of the tower. D 60% C To any convenient scale make AB (Fig. 69) the base of a right-angled triangle equal 100 units. Draw the line AC, making an angle of 60° with AB and intersecting BC at C. Then BC is the required height. 100-- FIG. 69. is found to be 60°. By calculation, but tan 60° =√√3; .. BC=100√3=173... ft. Ex. 2. From the top of a tower, the B height of which is 100√3 ft., the angle of depression of an object on a straight level road on a line with the base of the tower Find the distance of the object from the tower. In this case, drawing CD a horizontal line (Fig. 69) through C, the point of observation, making CB=100√3 ft. on any scale, and BA at right angles to CB. Then the point at which a line CA, drawn at an angle of 60° to the horizontal line DC, meets BA gives the distance BA required = 100 ft. EXERCISES. XLIX. 1. The angle of elevation of the top of a steeple is 45° from a point in the same horizontal plane as its base, and is 30° from a point 30 feet directly above the former point; find the height and distance of the steeple. 2. A pole is fixed on the top of a mound, and the angles of elevation of the bottom and top of the pole are 30° and 60° respectively; prove that the height of the pole is twice the height of the mound. 3. The altitude of a tower, observed at the end of a horizontal base of 100 yards measured from its foot, is 30°; find its height. 4. From the top of a tower 150 feet high the angles of depression of the top and bottom of a vertical column, standing on the same horizontal plane, were observed to be 22° 15′ and 44° 30'; find the height of the column. 5. A line AB, 450 yds. long, is measured close by the brink of a river, and a point C, close to the bank of the river on the other side, is observed both from A and B ; the angle CAB is 52°, and CBA is 70°; find the width of the river. 6. A tower on the bank of a river is 200 feet high, and carries a flag-staff 30 feet high. A man 6 feet in height stands at the bottom of the tower. To an observer on the opposite bank the man and the flag-staff subtend equal angles. Find the width of the river. 7. From the lower window of a house the angle of elevation of a church tower is observed to be 45°, and from a window 20 feet above the former, 40°; how far is the house from the church? 8. Looking down from the top of a hill, the foot of which is 125 feet below the level of his eye, a man sees a statue 40 feet high, and the column, 60 feet high, upon which it stands, subtending equal angles at his eye. How far off is he in a horizontal direction from the object? 9. What is the angle of depression of an object? From the top of a hill the angles of depression of two consecutive mile-stones on a straight, level road, were found to be 12° 13′ and 2° 45′ respectively; find the height of the hill. 10. The angle of elevation of the top of a hill is observed to be 5°; after walking one mile directly towards the hill the angle of elevation is 14° 30'. Find the height of the hill. 11. An object 10 ft. high is placed on the top of a tower, and subtends an angle of 6° at a place which is in the same horizontal plane as the foot of the tower, and is 50 ft. distant from it. Determine the height of the tower. 12. From the top of a tower by the seaside, 150 ft. high, it was found that the angle of depression of a ship's hull was 36° 18'. Find the distance of the ship from the foot of the tower. 13. ABC is a triangle with a right angle at C, CB is 30 feet long, and BAC is 20°. If CB be produced to a point P, such that PAC is 55°, calculate the length of CP. 14. The angular height of a tower is observed from two points A and B 1000 feet apart in the same horizontal line as the base of the tower. If the angle at A is 20° and at B 55°, find the height of the tower. 15. The angle of elevation of the top of a steeple is 30°. If I walk 50 yards nearer, the angle of elevation becomes 60°. What is the height of the steeple? 16. The elevation of a tower from a point A due N. of it is observed to be 45°, and from a point B due E. of it to be 30°. If AB=240 feet, find the height of the tower. 17. If from a point at the foot of the mountain, at which the elevation of the observatory on the top of Ben Nevis is 60°, a man walks 1900 feet up a slope of 30°, and then finds that the elevation of the observatory is 75°, show that the height of Ben Nevis is nearly 4500 feet. 18. The angular elevation of the top of a vertical pole changes from 45° to 30° as an observer moves 100 feet away from the pole in a horizontal line through the pole. Find the height of the pole, the observer's eye being 5 feet above the ground. 19. From a point P on the bank of a river just opposite a post Q on the other bank, a man walks at right angles to PQ to a point R, such that PR is 100 yards; he then observes the angle PRQ to be 32° 17'; find the breadth of the river, having given tan 32° 17′ = '6318. 20. A man wishes to find the height of a church spire which stands on a horizontal plane; at a point on this plane he finds the angle of elevation of the top of the spire to be 45°; after walking 100 feet toward the spire he finds the angle of elevation to be 60°; what is the height of the spire? 21. A and B are two points on one bank of a straight river and Ca point on the opposite bank; the angle BAC is 30°, the angle ABC is 60°, and the distance AB is 400 feet; find the breadth of the river. 22. Find the height of a chimney which is such that on walking towards it 100 feet in a horizontal line through its base, the angular elevation of its top changes from 30° to 45°. 23. The angles of elevation of a spire at two places due east of it and 200 feet apart are 45° and 30°; find the height of the spire. 24. A and B are two hill-tops 34,920 feet apart, and C is the top of a distant hill. The angles CAB and CBA are observed to be 61° 53′ and 76° 49′ respectively. Prove that the distance from A to C is 51,515 feet. 25. From the top of a cliff 1000 feet high, the angles of depression of two ships at sea are observed to be 45° and 30° respectively; if the line joining the ships points directly to the foot of the cliff, find the distance between the ships. Summary. All triangles consist of six parts, three angles and three sides; the sum of the three angles of a triangle is 180°. Right-angled triangles.—If one angle is a right angle or 90° the sum of the other two angles is 90°; each of these latter angles being less than 90° is called an acute angle, and in all right-angled triangles having the same acute angle the ratio of the sides is the same. These ratios are known as: The sine. The sine of an angle is formed by the ratio of the side opposite the angle to the hypothenuse. The cosine. The ratio of the side adjacent to the angle, to the hypothenuse. The tangent.-The ratio of the side opposite the angle to the side adjacent. The reciprocals form three other ratios. Thus the reciprocal of the sine is called the cosecant. The reciprocal of the cosine the secant. The reciprocal of the tangent the cotangent. When the acute angles are 60° and 30°, the sides of the triangle are proportional to 1, 2, and √3. When the acute angles are each 45°, sides are as 1 to √√2. The complement of an angle is the remainder after subtracting the angle from 90°. The supplement is the remainder after subtracting the angle from 180°. sin2A + cos2A=1; sec2A=1+tan2A ; CHAPTER XII. LABOUR-SAVING METHODS. THE SLIDE RULE. Slide Rule. It will be already clear to the reader who has followed the section dealing with logarithms, that by their use multiplication of two or more numbers is effected by adding the logarithms of the factors, and division by the subtraction of the logarithms of the factors. Or, shortly, by the use of logarithms multiplication is replaced by addition, and division by subtracHence if instead of the equal divisions of a scale (Fig. 70) unequal divisions corresponding to the logarithms were employed, then, when performed graphically, multiplication will correspond to addition and division to subtraction. tion. It is, as has been seen, an easy matter to add together two linear dimensions by means of an ordinary scale or rule. Thus, to add 2 and 3 units together. Assume the scale B (Fig. 70) to slide along the edge of the scale A, then the addition of the numbers 2 and 3 is made when the 2 on B is coincident with O on A; the addition of the two numbers is found to be 5 opposite the number 3 on the scale of A. If the scales on A and B were not divided in the proportion of the numbers, but of the logarithms of the numbers, |