CHAPTER V. THE ANALYSIS OF ANGULAR SECTIONS BY DEMOIVRE'S THEOREM. 115. THE preceding chapters comprise what is properly regarded as the Elements of Trigonometry, and in them will be found the Demonstrations and Applications of all the Principles and Theorems employed in a course of Elementary Mathematical Science. We proceed now to a series of propositions established upon Analytical grounds, which at the same time that they are indispensably necessary in the Investigations of Physical Science, present to our notice many of the results already obtained, as well as several remarkable conclusions which it would be otherwise difficult to arrive at. 116. To find expressions for all angles, whose sines, coines, &c. are equal to sin 0, cos 0, &c., respectively. The reasoning used in (7) of Article (35) gives sin 0 = ± sin (2rπ ± 0) = ± sin {(2r + 1) π = 0} : : cos = + cos (2rπ ±0) cos {(2r + 1) π ± 0} : tan 0 = ±tan (2rπ ± 0) = ±tan {(2r + 1) π ± 0} : &c. where r may be any whole number whatever: in the first and last of which, the double sign ±, may be affixed to the angles themselves instead of their functions, and in the second it may be affixed to the angles by leaving the signs of the functions as they are, according to Article (31). Hence, all the functions of the angular magnitude 2rπ0 will be equal in every respect to those of the angle: whereas the functions of (2r+1) π±0 may differ from them in the algebraical sign: and admitting that may be either a positive or negative quantity, the same circumstances will be found to occur in the following articles. cessive integral values are assigned to r. If u with the figures 0, 1, 2, 3, &c. suffixed, denote the values of cos , n corresponding to the values 0, 1, 2, 3, &c. of r respectively, we shall have from which it appears there are n, and only n different 2rπ + 0 values of cos &c. n n , corresponding to the values 0, 1, 2, 1, of r: inasmuch as the same values afterwards recur in the same order: and the function is therefore said to be periodical. The values above found are in general all different from each other : for, if r' and r" be any two values of r, 2 (r"-r') 1 π n , unless n be a multiple of 2π, which is inadmissible, because r', r" and .. r"- r'<n: and although equal values of the function may be obtained by assigning particular values to 0, the number of its values will still If r be greater than n 1, it has been seen that multiples of 2π produce no alteration in the values of the proposed function. be n. will also be found by putting 0, 1, 2, 3, &c. n − 1, for r: and if 0 = 0, the same will therefore hold good for 118. If m be any rational quantity whatever, either integral or fractional, positive or negative: {cos 0=√1 sin 0}" = cos me±√√-1 sin mə. (1) Let the index be a positive whole number m: then {cos 0±√- 1 sin 0}2 = {cos 0 ± √— 1 sin 0} ㄓ = cos 30 ± √1 sin 30: and so on: hence, if we assume the law observed here, to hold good for the index m - 1, so that ㄓ {cos 0 + √-1 sin 0-1 = cos (m − 1) 0 ± √√1 sin (m − 1) 0, we shall have {cos 0-1 sin 03 = {cos (m − 1) 0 ± √-1 sin (m −1) 0} {cos 0 ± √-1 sin 0} = = cos (m-1) cos 0 - sin (m − 1) 0 sin 0 - ± √− 1 {sin (m − 1) 0 cos 0 + cos (m −1)0 sin 0} or, if the formula be true for any one value of the index, it is proved to be true for the next superior value: but it has been demonstrated in the cases when the index is 2 and 3, and therefore it is true when the index is 4, 5, &c.: that is, it is generally true that + {cos 0±√-1 sin 0} = cos me√√-1 sin me, when the index m is a positive whole number. (2) Let the index be a negative whole number - m : then F = cos m0 = √− 1 sin me, by the preceding case, ㄓ = cos (− m0) ± √—I sin (- m☺), by Article (31). (3) Let the index be a fraction, either positive or ± {cos 0-1 sin 0) = cos m0√-1 sin mo 119. COR. 1. Hence, when m is a positive whole number, by Article (24), we shall have = cos (2mrπ + m3) ± √− 1 sin (2mrπ + m0) = cos me± that is, each member of the equality has only one value, whatever whole number m may be. If however the index be fractional, as m - n {cos (2rπ+0)√-1 sin (2rπ+0)}" ; then will whereof the latter member has n different values corre sponding to the values 0, 1, 2, &c. n-1, of r, as seen in Article (117): and these n values are the n roots of the expression {cos 0-1 sin 0m: that is, each member of the equality has the same number of different values, which are merely indicated in the former, but may be actually exhibited in the latter, by assigning to r the values 0, 1, 2, 3, &c. n - 1. |