XIX. Oh the Values of the Sine and Cosine of an Infinite Angle. By 

 S. Earnshaw, M.A., of St. John's College, Cambridge. 



[Read December 9, 1844.] 



The usage of Mathematicians in reference to the symbols Sin co and Cos os does not seem to be 

 in accordance with their expressed opinions. It does not appear to be questioned eitlier by English 

 or Foreign writers, that when ,i' becomes infinite Sin a? and Cos x cannot be said to be in one part 

 of their periodicity rather than another. If this mean any thing, it must be understood to signify 

 that Sin eo and Cos os are indefinite. Yet this is not borne out in the usuage of these symbols 

 which we find in the writings of any author. Indeed, an opinion has been expressed that their 

 indeterminateness is only apparent, and therefore not real : and that analysis has furnished definite 

 equivalents for them by legitimate processes of investigation on principles which are allowed : and 

 though some writers on Definite Integrals have abstained from stating in direct terms what are the 

 values which analysis assigns to Sin eo and Cos oo, all agree in practically affirming " that both the 

 Sine and Cosine of an infinite angle are equal to zero." But while we find these values used where- 

 ever Sin 05 and Cos m occur in investigations, we do occasionally meet with expressions of doubtful- 

 ness respecting their universal truth. This seems to indicate that in the opinion of such writers 

 the values of Sin os and Cos eo depend on the circumstances under which they occur; but what those 

 circumstances are which have this power over Sin co and Cos os I do not find any where pointed 

 out. In fact, upon tracing the origin of this doubt respecting the universal truth of the equations 

 Sin OS = 0, Cos OS = 0, I find that it has arisen from the occurrence of certain results of a character 

 so obviously suspicious, perhaps I might say, erroneous and contradictory of evident truths, as to 

 create a reasonable doubt of the propriety of writing zero for Sin os and Cos os in those cases. 

 But though results have thus forced some writers to doubt respecting the general truth of the 

 equations Sin os = and Cos eo = 0, it does not appear that they have any where admitted the 

 demonstrations of the truth of these equations to be defective. We find ourselves then in this 

 difficult position ; — we have certain investigations presented to us in which there occur no doubted 

 steps, and these investigations present us with certain absolute results ; — but the certainty of these 

 results thus established by a process of mathematical reasoning, the accuracy of which is no where 

 called in question, we are afterwards required to look upon with suspicion ; — and that sort of 

 suspicion which while it throws doubt upon every thing affords us no clue for ascertaining what are 

 the cases to which alone it ought to be attached. It is obviously desirable that some effort should 

 be made to remove this uncertainty. Now some light may Ije thrown upon this difficulty by 



27r 

 considering that Sin«.r and Cosji.r go through a whole period of values while *• increases by — . 



2ir . . 

 As long as n is finite — is finite, and all tlie values included in a period are therefore consecutive, 

 n 



But what happens when n increases in value ? We easily see that as n increases the whole period 



becomes condensed so as to occupy a shorter and shorter portion of the current variable ; and 



that when n approaches os, the values are no longer consecutive but simultaneous: — hence as n 



increases towards os a whole period of values of Sin* or Cosx tends to become simultaneous, and 



in the limits are simultaneous.- i.e., Sin eo has at once all values from — 1 to + 1 ; and the same 



