a number "much smaller than unity, and whose logarithm would consequently be negative. When multiplied by 10" it becomes =2908882. a number whose logarithm is 6.4637261, and consequently we find in our tables, log. sin. 1' 6.4637261. A table constituted upon this principle is called a Table of Logarithmic Sines, Cosines, Tangents, &c. and by this nearly all the practical operations of trigonometry are usually performed. It is manifest, from these remarks, that before we can apply formulæ deduced in the preceding chapters to practical purposes, we must transform them in such a manner as to render the several trigonometrical quantities identical with those registered in our tables. The sines, cosines, &c. we have hitherto employed, are called Trigonometrical qnantities calculated to a radius unity; those registered in the tables, Trigonometrical quantities calculated to radius R. The problem to be solved therefore is To transform an expression calculated to a radius unity, to another calculated to a radius R Let us represent sin. to radius unity by m. R by n. and so for all the other trigonometrical quantities. Hence, in order to transform an expression calculated to radius unity, to another calculated to radius R, we must divide each of the trigonometrical quantities by R. If any of the trigonometrical quantities enter in the square, cube, &c. these must of course be divided by R2 R3, &c. As observed above, R may be any given number whatever, the number usually employed in the ordinary tables being 10", and therefore log. R = 10 Take as an example such an expression as a sin. = b tan.2 o in order to reduce this to an expression which we can compute by our tables we must, according to the above rule, divide each of the trigonometrical quantities by the proper power of R: the expression then becomes log. a + log. R + log. sin. = log. b + 2 log. tan. an expression which may be calculated by the tables. If the expression calculated to radius unity be of the form it requires no modification, for if we divide both terms of the fraction sin. sin. by R, we shall not alter its value. We need not prosecute this subject farther, as numerous examples of these transformations will occur at every step in the succeeding chapters. CHAPTER VI. ON THE SOLUTION OF RIGHT-ANGLED TRIANGLES. Every plane triangle being considered to consist of 6 parts, the three sides, and the three angles, if any of these three parts be given, we can, in general, determine the remaining parts by trigonometry. In right-angled triangles, the right angle is always known, and therefore any two other parts being given, we can, in general, determine the rest. We shall thus have five different cases. 1. When one of the acute angles and the hypothenuse is given. 2. When one of the acute angles and a side is given. 3. When the hypothenuse and one side is given. 4. When the two sides are given. 5. When the two acute angles are given. Let ABC be a right-angled triangle, C the right ngle. Let the sides opposite to the angles A and B be denoted respectively by a, b, and let the hypothenuse be called c. Case 1, Given A, c, required B, a. b. A+B=90° A B B=90°-A whence B is known - - -(1) By Chap. III. prop. 1, a=c sin. A Adapting this expression to computation by the tables. log. a=c sin. A R .log. a=log. c+log. sin.A-log.R, whence a is known - (2) In like manner, b=c cos. A log.blog.c+log. cos. A-log. R, whence b is known (3) If B, c are given, and A, a, b required, we shall have precisely in the same manner, A=90°- B · (4) (5) (6) log.blog. a+log. cot. A-log. R, whence b is known (8) Again, a=c sin. A ..log. c=log. R+log. a-log. sin. A, whence c is known (9) If A, b, be given, B, a, c, we shall have in like manner, B=90°-A Case 3. Let a, c be given, required b, A, B. b2=c2- a2 =(c+a) (c-a) .. 2 log. b=log. (c+a)+log. (c—a) whence b is known (19) ..log. sin. A=log. R+log. a-log. c, whence A is known, (20) So also, cos. B α ..log. cos. B=log. R+log. a-log. c, whence B is known, (21) If b, c be given, and a, A, B required, we shall have 2 log. a=log. (c+b)+log. (c—b) log. cos. A=log. R+log b-log. c log. sin. B=log. R+log. b-log. c Case 4. Let a, b be given, required A, B, c. (22) (23) (24) ...log.tan. A=log. R+log. a-log. b, whence A is known, (25) So also, ... log. tan. B=log. R+log. b—log. a, whence B is known, (26) c=a+b2, whence c is known, Case 5. Given A, B, required a, b, c. (27) It is manifest that this case does not admit of a solution, for any number of unequal similar triangles may be constructed, having their angles equal to the angles A, B, C. We shall conclude this chapter by giving some numerical examples. Example 1. Given A=26° 41' 6", c=6539.76 yards, required a. The number in the tables corresponding to the logarithm 3.4678904 is found to be 2936.91. a=2936.91 yards. In like manner, the side b may be determined, if required. Example 2. Given c=6539.76 yards, a=2936.91 yards, required b, A, B. By (19). 2 log. blog. (c+a)+log. (ca) c+a=9476.67 c-a-3602.85 The number in the tables corresponding to the logarithm 3.7666509 is 5843.2 On referring to our tables, we shall find that the angles whose logarithmic sine is 9.6523286 is 26° 41' 6", which is consequently the value of A. A being known, B is determined at once by subtracting the value of A from 90°, or B may be determined independently of A by (21). log. cos. B=log. R+log. a-log. c. |