30. Given an Acute Angle and the Hypotenuse. For example, given A = 43° 17', c = 26, find B, a, and b. As usual, when a four-place table is employed, the result is given to four figures only. The check is left for the student. 31. Given an Acute Angle and the Opposite Side. For example, given A = 13° 58', a 15.2, find B, b, and c. = In dividing 15.2 by 0.2414, we adopt the modern plan of first multiplying each by 10,000. Only part of the actual division is shown. Instead of dividing a by sin A to find c, we might multiply a by csc A, as on page 22, except that tables do not generally give the cosecants. It will be seen in Chapter III that, by the aid of logarithms, we can divide by sin A as readily as multiply by csc A, and this is why the tables omit the cosecant. 32. Given an Acute Angle and the Adjacent Side. For example, given A = 27° 12', b = 31, find B, a, and c. We might multiply b by sec A instead of dividing by cos A. The reason for not doing so is the same as that given in § 31 for not multiplying by csc A. 33. Given the Hypotenuse and a Side. For example, given a = 47, C = 63, find A, B, and b. B α=47 In the case of √c2 - a2 we can, of course, square c, square a, take the difference of these squares, and then extract the square root. It is, however, easier to proceed by factoring c2 - a2 as shown. This will be even more apparent when we come, in Chapter III, to the short methods of computing by logarithms. 34. Given the two Sides. For example, given a = 40, b = 27, find Of course c can be found in other ways. For example, after finding tan A we can find A, and hence can find sin A. Then, because sin A = a/c, we have ca/sin A. When the numbers are small, however, it is easy to find c from the relation given above. = 35. Checks. As already stated, always apply some check to the results. For example, in § 34, we see at once that a2 1600 and b2 is less than 302, or 900, so that c2 is less than 2500, and c is less than 50. Hence the result as given, 48.26, is probably correct. We can also find B independently. Solve the right triangle ACB, in which C = 90°, given : 33. Each equal side of an isosceles triangle is 16 in., and one of the equal angles is 24°10'. What is the length of the base? 34. Each equal side of an isosceles triangle is 25 in., and the vertical angle is 36° 40'. What is the altitude of the triangle? 35. Each equal side of an isosceles triangle is 25 in., and one of the equal angles is 32° 20' 30". What is the length of the base? 36. Each equal side of an isosceles triangle is 60 in., and the vertical angle is 50° 30' 30". What is the altitude of the triangle? 37. Find the altitude of an equilateral triangle of which the side is 50 in. Show three methods of finding the altitude. 38. What is the side of an equilateral triangle of which the altitude is 52 in.? = 39. In planning a truss for a bridge it is necessary to have the upright BC 12 ft., and the horizontal AC 8 ft., as shown in the figure. What angle does AB make with AC? with BC? 40. In Ex. 39 what are the angles if AB 12 ft. and AC= 9 ft.? 41. In the figure of Ex. 39, what is the length of BC if AB=15 ft. and x = 62° 10'? 42. Two angles of a triangle are 42° 17' and 47° 43' respectively, and the included side is 25 in. Find the other two sides. = 10 ft., 43. A tangent AB, drawn from a point A to a circle, makes an angle of 51° 10' with a line from A through the center. If AB what is the length of the radius? 44. How far from the center of a circle of radius 12 in. will a tangent meet a diameter with which it makes an angle of 10° 20'? 45. Two circles of radii 10 in. and 14 in. are externally tangent. What angle does their line of centers make with their common exterior tangent? CHAPTER III LOGARITHMS 36. Importance of Logarithms. It has already been seen that the trigonometric functions are, in general, incommensurable with unity. Hence they contain decimal fractions of an infinite number of places. Even if we express these fractions only to four or five decimal places, the labor of multiplying and dividing by them is considerable. For this reason numerous devices have appeared for simplifying this work. Among these devices are various calculating machines, but none of these can easily be carried about and they are too expensive for general use. There is also the slide rule, an inexpensive instrument for approximate multiplication and division, but for trigonometric work this is not of particular value because the tables must be at hand even when the slide rule is used. The most practical device for the purpose was invented early in the seventeenth century and the credit is chiefly due to John Napier, a Scotchman, whose tables appeared in 1614. These tables, afterwards much improved by Henry Briggs, a contemporary of Napier, are known as tables of logarithms, and by their use the operation of multiplication is reduced to that of addition; that of division is reduced to subtraction; raising to any power is reduced to one multiplication; and the extracting of any root is reduced to a single division. For the ordinary purposes of trigonometry the tables of functions used in Chapter II are fairly satisfactory, the time required for most of the operations not being unreasonable. But when a problem is met which requires a large amount of computation, the tables of natural functions, as they are called, to distinguish them from the tables of logarithmic functions, are not convenient. For example, we shall see that the product of 2.417, 3.426, 517.4, and 91.63 can be found from a table by adding four numbers which the table gives. 4.27 36.1 5176 we shall see that the result can be found In the case of X from a table by adding six numbers. Taking a more difficult case, like that of is necessary merely to take one third of the sum of four numbers, after which the table gives us the result. |