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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.
106. tun, tan . + tun tun ) + tun tun = 1.
Prove the following statements when we take for sin-, cos-, etc., their least positive value.
LOGARITHMS AND LOGARITHMIO TABLES. — TRIGO
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.
.:: 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.
- 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
... m = 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
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.