Prove the following statements, A, B, C being the angles of a triangle. 103. sin A cos A – sin B cos B + sin C cos C = 2 cos A sin B cos C. 104. cos 2 A + cos2 B + cos2C 1-4cos A cos B cos C. 105. sin? A – sin’B + sin’C = 2 sin A cos B sin C. C С А B 106. tun, tan . + tun tun ) + tun tun = 1. B A + 2 2 Prove the following statements when we take for sin-, cos-, etc., their least positive value. CHAPTER IV. LOGARITHMS AND LOGARITHMIO 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 x = logam, the base being put as a suffix. * The relation between the base, logarithm, and number is expressed by the equation, (base) log = number. 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) loga at = x, and conversely (2) alogam = m. Taking the logarithms of both sides of the equation ax = m, we have loga a= x=log m. Conversely, taking the exponentials of both sides of x = logam to base a, we have a = aloga m=m. at = m and x = logam 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. (3) The logarithm of zero in any system whose base is greater than 1 is minus infinity. 1 1 For a-= 0; therefore log 0 ao (4) The logarithm of a product is equal to the sum of the logarithms of its factors. For let - logam, and y= logan. = a .:: logamn = x+y= logam + logan. Similarly, log, mnp= logam + logan + log.p, and so on for any number of factors. Thus, log 60 = log (3 x 4 x 5), = log 3 + log 4 + log 5. (5) The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. For let x = logam, and y= logan. Thus, 17 log - log 17 – log 5. (6) The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power. For let - logam. ... m = a. X= a .. log, MP = px = plogam. (7) The logarithm of any root of a number is equal to the logarithm of the number divided by the index of the root. For let x = logam. .. m = a". = 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 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 : log10371.49 = 2.56995 0.85082 which is the logarithm of 7.092752, the quotient required. |