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Prove the following statements, A, B, C being the angles of a triangle.

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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
2

A

+ 2

2

Prove the following statements when we take for sin-, cos-, etc., their least positive value.

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111. tan- V5(2 – V3) – cot-? V5 (2 + V3)= cot-1 V5.

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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.
For ao = 1, whatever a may be; therefore log 1 = 0.
(2) The logarithm of the base of any system is unity.
For a'= a, whatever a may be; therefore loga a = = 1.

(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.

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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.

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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=

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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".

=

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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
log10357 = 2.55267

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 :

log10371.49 = 2.56995
log10 52.376 = 1.71913

0.85082 which is the logarithm of 7.092752, the quotient required.

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