Take BD (fig. Art. 12) = x, then AD=x/3; with centre B, distance AD describe a circle meeting AD in E, join BE, then and the angle BED has been found as required. 3. If a line of 10 feet be represented by 3, what is the unit of length? 4. How is 37 yards, 2 feet, 8 inches represented, when the unit of length is 4 inches? Ans. By 341. 5. If the angle of an equilateral triangle be taken as the unit of measurement, how many degrees will be contained in the angle .3 ? 6. Find the number of grades in 27°.35'. 24". 7. What is the supplement of 227o, 35`,46“? 8. The complement of 45° ± a is 45° F a. Ans. 30°.65. 9. The difference of two acute angles of a right-angled triangle is do, find the angles. 10. The vertical angle of an isosceles triangle is ao, find the angles at the base. 11. The length of a person's shadow is twice his height, find the altitude of the sun, having given tan 26°.34′ = .5. 12. Shew that (sin 60°— sin 45°) (cos 30° + cos 45°) = sin330o. 13. Shew that (sin 30°+ cos 30°) (sin 60° — cos 60°) = cos 60o. sin 45° - sin 30° 14. Shew that =(sec 45°-tan 45°). sin 45° + sin 30° 15. Determine the height of an object whose elevation is 40°, the observer being 140 feet distant, and his eye 5 feet from the ground, having given that tan 40°.85. Ans. 124 feet. 16. Construct the angle whose tangent is √2. 17. Construct the angle whose sine is √2-1, 2 18. A ladder of length 7 is placed against a wall so that the angle it makes with the ground is double that which it makes with the wall: find how far from the wall the foot of the ladder is. Ans. 2° 19. The town C is halfway between the towns D and E; and the towns C, E, F are equidistant from each other. If distance from D to E is 12 miles, find the distance from D to F. Ans. 63 miles. 20. A ladder 20 feet long just reaches the top of a wall when its foot is 13 feet from the foot of the wall; shew that when its foot is 5 feet from the wall, the ladder projects 4 feet beyond the top of the wall. 21. In a right-angled triangle the lengths of lines drawn from the acute angles to the middle points of the opposite sides are d, d' respectively. Find the sides of the triangle. 2 Ans. √ (4ď2 — d12), 2 √/15 √ (4d12 — d2). 22. A person travelling southwards observes two objects towards the S.E. After 8 miles travelling, one of them is N.E., and the other E. Their distances from him are then 4/2 miles, and 8 miles respectively. 23. Whilst sailing due West, I observe two ships at anchor directly North of me: after sailing 6 miles the directions of the ships make angles 60°, 30° with my course respectively. The distance between them is 4√3 miles. 24. A staff 1 foot long stands on the top of a tower 200 feet high. Shew that the angle it subtends at a place 100 feet from the foot of the tower is 6'. Given tan 63°.27' 2, tan 63°. 33'2.01. = 25. The angles of depression of the top and bottom of a column observed from a tower 108 feet high are 30°, 60° respectively. Shew that the height of the column is 72 feet. 26. From the top of a column 100 feet high, the angles of depression of two objects in a line with the column, and in the horizontal plane on which the column stands, are observed to be 30° and 60°. The distance between them 200 is feet. √/3 27. The length of a road in which the ascent is 1 foot in 5 from the foot of a hill to the top is a mile and two-thirds: what will be the length of a zigzag road in which the ascent is 1 foot in 12? Ans. 4 miles. EXAMPLES. (B). 1. The same line is represented by m and n in two systems of measurement, compare the units of length in the two systems. 2. What is the greatest unit of length with which .625 feet may be represented by represented by an integer? 3. Find the number of French minutes in one English minute, and the number of French seconds in one English second. Ans. 1.851, 3.086419753. 4. The angles of a triangle are in A. P; shew that one of them must equal 60°. 5. The supplement of one angle of a triangle is double the complement of another, and triple that of the third find the angles. Ans. 81, 4019, 57 degrees. 6. A person observes the angular elevation of a column; after approaching a feet, he finds its elevation doubled; again approaching b feet it is again doubled. Shew that the second point of observation is feet from the foot of the tower. 26 7. Construct the angle whose tangent is √5 – 1. 8. At the foot of a mountain the elevation of its summit is 45°. After ascending 1 mile up a slope of 30° its elevation is found to be 60°. The height of the mountain is 3+1 miles. 9. A person at the edge of a river observes the elevation a of a tower on the other side; on retreating a feet he finds a tan a tan B its elevation to be B: the height of the tower is tan a-tan ẞ' 10. A rock is observed from a ship to bear N. N.W; after sailing 10 miles in direction E.N.E the rock is due West: its distance from the ship at the first observation was 10 (√2—1) miles. Given tan 22°/2-1. 11. Two persons A and B start from two points distant 400 yards. B starts at right angles to the line joining the two points at the rate of 90 yards a minute. A starts in a direction to catch B as soon as possible at the rate of 150 yards a minute; find how long he will be before he catches him, and the direction in which he will walk, having given sin 36°. 53' = Ans. 3 minutes, 20 seconds. 3 = 12. A tower is situated at the top of a hill whose inclination is 30°. The angle subtended by the tower at the foot of the hill is 15°; and on walking a yards up the hill it is found to be 30o. The height of the tower is α √3 feet. 13. From two points in the diameter of a circle produced tangents are drawn to the circle. Given the distance c between the points, and the inclinations a, ẞ of the tangents to the diameter, shew that the radius of the circle с cosec Bcosec a 14. A person travelling along a road observes the elevation a of a tower the nearest distance (a) of which from the road is known; at the same time he observes the angular distance ẞ of the top of the tower from an object in the road. sin α The height of the tower is a √(cos2 a cos2 B)* 15. At three positions in the same horizontal plane distant from each other 60, 80, 100 feet respectively, the elevation of a tower is observed to be 45°; find its height. Ans. 50 feet. 16. Two spectators at two stations, distant 2a from each other, observe the elevation of a kite to be a at each station, and the angle subtended by the kite and the other station to be B; shew that the height of the kite is a sec ẞ sin a. 17. A person standing on the top A of a light-house AB, of known height 300 feet, observes a ship sailing from C to D in a straight line: he knows that CD is perpendicular to the plane ACB, and he observes the angle CAD=30°, BAD=60°. Shew that CD=300 feet; BD = 300/3 feet. 18. The altitude of the sun is 45°, and it is at a point of the compass 60° from the south. The breadth of the shadow of a south wall is one-half its height. 19. The angular altitude and breadth of a cylindrical tower are observed to be a and ẞ respectively; but at a point a feet nearer the foot of the tower they are a', B' respectively: find the height and radius of the tower. cosec cosec B. T. 2 |