49. Find the value of F'in terms of c and A. a= 10; solve the triangle. solve the triangle. -55. Given F= 12, A=29°; solve the triangle. solve the triangle. 56. Given F 100, c= 22; 57. Find the angles of a right triangle if the hypotenuse is equal to three times one of the legs. 58. Find the legs of a right triangle if the hypotenuse = 6, and one angle is twice the other. 59. In a right triangle given c, and AnB; find a and b. 60. In a right triangle the difference between the hypotenuse and the greater leg is equal to the difference between the two legs; find the angles. The angle of elevation of an object (or angle of depression, if the object is below the level of the observer) is the angle which a line from the eye to the object makes with a horizontal line in the same vertical plane. 61. At a horizontal distance of 120 feet from the foot of a steeple, the angle of elevation of the top was found to be 60°30'; find the height of the steeple. 62. From the top of a rock that rises vertically 326 feet out of the water, the angle of depression of a boat was found to be 24°; find the distance of the boat from the foot of the rock. 63. How far is a monument, in a level plain, from the eye, if the height of the monument is 200 feet and the angle of elevation of the top 3° 30'? 64. In order to find the breadth of a river a distance AB was measured along the bank, the point A being directly opposite a tree C on the other side. The angle ABC was also measured. If AB=96 feet, and ABC= 21° 14′, find the breadth of the river. If ABC= 45°, what would be the breadth of the river? 65. Find the angle of elevation of the sun when a tower a feet high casts a horizontal shadow b feet long. Find the angle when a 120, b = 70. = 66. How high is a tree that casts a horizontal shadow b feet in length when the angle of elevation of the sun is 4°? Find the height of the tree when b= 80, A = 50°. 67. What is the angle of elevation of an inclined plane if it rises 1 foot in a horizontal distance of 40 feet? 68. A ship is sailing due north-east with a velocity of 10 miles an hour. Find the rate at which she is moving due north, and also due east. . 69. In front of a window 20 feet high is a flower-bed 6 feet wide. How long must a ladder be to reach from the edge of the bed to the window? 70. A ladder 40 feet long may be so placed that it will reach a window 33 feet high on one side of the street, and by turning it over without moving its foot it will reach a window 21 feet high on the other side. Find the breadth of the street. 71. From the top of a hill the angles of depression of two successive milestones, on a straight level road leading to the hill, are observed to be 5° and 15°. Find the height of the hill. 72. A fort stands on a horizontal plain. The angle of elevation at a certain point on the plain is 30°, and at a point 100 feet nearer the fort it is 45°. How high is the fort? 73. From a certain point on the ground the angles of elevation of the belfry of a church and of the top of the steeple were found to be 40° and 51° respectively. From a point 300 feet farther off, on a horizontal line, the angle of elevation of the top of the steeple is found to be 33° 45'. Find the distance from the belfry to the top of the steeple. 74. The angle of elevation of the top of an inaccessible fort C, observed from a point A, is 12°. At a point B, 219 feet from A and on a line AB perpendicular to AC, the angle ABC is 61° 45'. Find the height of the fort. § 17. THE ISOSCELES TRIANGLE. An isosceles triangle is divided by the perpendicular from the vertex to the base into two equal right triangles. Therefore, an isosceles triangle is determined by any two parts that determine one of these right triangles. Let the parts of an isosceles triangle ABC (Fig. 13), among which the altitude CD is to be in 2. C+2A с a 2 a the altitude. A = one of the equal angles. C= the angle at the vertex. For example: Given a and c; required A, C, h. 180°; .. C 180° - 2 A = 2(90° — A). 3. h may be found directly in terms of a and c from the equation which gives h2 + a2, c2 4 h = √(a — 1⁄2 c) (a + 1⁄2 c). But it is better to find the angles first, and then find h from either one of the two equations, The numerical values of A, C, and h may be computed by the aid of logarithms, as in the case of the right triangle. The area F of the triangle may be found, when e and h are given or have been found, by means of the formula F= ch. 17. Find the value of Fin terms of a and c. 18. Find the value of Fin terms of a and C. 19. Find the value of Fin terms of a and A. 20. Find the value of Fin terms of h and C. 21. A barn is 40 × 80 feet, the pitch of the roof is 45°; find the length of the rafters and the area of both sides of the roof. 22. In a unit circle what is the length of the chord corresponding to the angle 45° at the centre? 23. If the radius of a circle 30, and the length of a chord 44, find the angle at the centre. 24. Find the radius of a circle if a chord whose length is 5 subtends at the centre an angle of 133°. 25. What is the angle at the centre of a circle if the corresponding chord is equal to of the radius? 26. Find the area of a circular sector if the radius of the circle = = 12, and the angle of the sector = 30°. § 18. THE REGULAR POLYGON. Lines drawn from the centre of a regular polygon (Fig. 14) to the vertices are radii of the circumscribed circle; and lines drawn from the centre to the middle points of the sides are radii of the inscribed circle. These lines divide the polygon into equal right triangles. Therefore, a regular polygon is determined by a right triangle whose sides are the radius of the circumscribed circle, the radius of the inscribed circle, and half of one side of the polygon. If the polygon has n sides, the angle of this right triangle at the centre is equal to 1/360° 180° If, also, a side of the polygon, or one of the above-mentioned radii, is given, this triangle may be solved, and the solution gives the unknown parts of the polygon. |