Prove the following statements, A, B, C being the angles 103. sin A cos A — sin B cos B + sin C cos C =2 cos A sin B cos C. 104. cos 2 A+ cos 2 B + cos 2 C = =- 1 4 cos A cos B cos C. 105. sin2 A sin2 B + sin2C 2 sin A cos B sin C. H Prove the following statements when we take for sin-1, cos', etc., their least positive value. 108. 2 tan-1(cos 20) = tan-1 (cot20 — tan2 2 CHAPTER IV. LOGARITHMS AND LOGARITHMIC TABLES. TRIGO NOMETRIC TABLES. 62. Nature and Use of Logarithms. The numerical calculations which occur in Trigonometry are very much abbreviated by the aid of logarithms; and thus it is necessary to explain the nature and use of logarithms, and the manner of calculating them. The logarithm of a number to a given base is the exponent of the power to which the base must be raised to give the number. = Thus, if a m, x is called the "logarithm of m to the base a," and is usually written a log, m, the base being put as a suffix.* = The relation between the base, logarithm, and number is expressed by the equation, Thus, if the base of a system of logarithms is 2, then 3 is the logarithm of the number 8, because 23 = 8. If the base be 5, then 3 is the logarithm of 125, because 53 = 125. 63. Properties of Logarithms. The use of logarithms depends on the following properties which are true for all logarithms, whatever may be the base. * From the definition it follows that (1) log, ax = x, and conversely (2) alogam = m. Taking the logarithms of both sides of the equation a*: m, we have loga a* = x=log m. Conversely, taking the exponentials of both sides of x = logam to base a, we have ax = alogam = m. ax = m and x = log, m are thus seen to be equivalent, and to express the same relation between a number, m, and its logarithm, x, to base a. (1) The logarithm of 1 is zero. For ao = 1, whatever a may be; therefore log 1 = 0. (2) The logarithm of the base of any system is unity. For a1=a, whatever a may be; therefore log, a = 1. (3) The logarithm of zero in any system whose base is greater than 1 is minus infinity. (4) The logarithm of a product is equal to the sum of the logarithms of its factors. (5) The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. (6) The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power. (7) The logarithm of any root of a number is equal to the logarithm of the number divided by the index of the root. It follows from these propositions that by means of logarithms, the operations of multiplication and division are changed into those of addition and subtraction; and the operations of involution and evolution are changed into those of multiplication and division. 1. Suppose, for instance, it is required to find the product of 246 and 357; we add the logarithms of the factors, and the sum is the logarithm of the product: thus, log10 246 = 2.39093 log103572.55267 4.94360 which is the logarithm of 87822, the product required. 2. If we are required to divide 371.49 by 52.376, we proceed thus: which is the logarithm of 7.092752, the quotient required. |