EXAMPLES. 108. 1. Given c = 203.76, B= 21° 43'. Find a and b. In this case the formula to be used are 2. Given a = 13, A= 67° 7'. Find b and c. Note. It is customary in practice to omit writing the 10 after the mantissa of a negative logarithm, as illustrated in Ex. 3. 109. In the Trigonometrical solution of a triangle by the method of Case II., it is necessary to first find one of the angles, and the remaining side may then be calculated. It is possible however to obtain the third side directly, without first finding the angle, by Geometrical methods. Thus in Ex. 3, Art. 108, we have by Geometry, Whence, By logarithms, a2 + b2 = c2: Vc2 c2b2=√(c+b) (c-b). log a = [log (c + b) + log (c—b)]. c+b= .4593; log 9.6621 - 10 = = If the given sides are a and b the formula for c is Va+b2, which is not adapted to logarithmic computation. In such a case it is usually shorter to proceed according to the rule of Art. 107. Solve the following isosceles triangles, in which A and B are the equal angles, and a, b, and c denote the sides opposite the angles A, B, and C, respectively: 37. A regular pentagon is inscribed in a circle whose diameter is 24 inches. Find the length of its side. 38. At a distance of 100 feet from the base of a tower, the angle of elevation of its top is observed to be 38°. Find its height. 39. What is the angle of elevation of the sun when a tower whose height is 103.7 feet, casts a shadow 167.3 feet in length? 40. If the diameter of a circle is 3268, find the angle at the centre subtended by an arc whose chord is 1027. 41. If the diameter of the earth is 7912 miles, what is the distance of the remotest point of the surface visible from the summit of a mountain 14 miles in height? 42. Find the length of the diagonal of a regular pentagon whose side is 7.028 inches. 43. What is the angle of elevation of a mountain-slope which rises 238 feet in a horizontal distance of one-eighth of a mile? 44. From the top of a lighthouse, 133 feet above the sea, the angle of depression of a buoy is observed to be 18° 25'. Required the horizontal distance of the buoy. 45. A ship is sailing due east at the rate of 7.8 miles an hour. A headland is observed to bear due north at 10.37 A.M., and 33° west of north at 12.43 P.M. Find the distance of the headland from each point of observation. 46. If a chord of 41.36 feet subtends an arc of 145° 37', what is the radius of the circle? 47. The length of the side of a regular octagon is 12 inches. Find the radii of the inscribed and circumscribed circles. 48. How far from the foot of a pole 80 feet high must an observer stand, so that the angle of elevation of the top of the pole may be 10°? 49. If the diagonal of a regular pentagon is 32.83 inches, what is the radius of the circumscribed circle? 50. From the top of a tower, the angle of depression of the extremity of a horizontal base line, 1000 feet in length measured from the foot of the tower, is observed to be 21° 16' 37". Find the height of the tower. 51. If the radius of a circle is 723.29, what is the length of the chord which subtends an arc of 35° 13'? 52. A regular hexagon is circumscribed about a circle whose diameter is 10 inches. Find the length of its side. 53. From the top of a lighthouse, 200 feet above the sea, the angles of depression of two boats in line with the lighthouse are observed to be 14° and 32° respectively. What is the distance between the boats? 54. A ship is sailing due east at a uniform rate of speed. At 7 A.M., a lighthouse is observed bearing due north, 10.32 miles distant, and at 7.30 A.M. it bears 18° 13' west of north. Find the rate of sailing of the ship and the bearing of the lighthouse at 10 A.M. |