Trigonometrical Functions* 1 99 



sin x, cos x, &c, in terms of x are true in all cases, yet the de- 

 velopments of the inverse functions, which, a priori, we should 

 expect to possess equal generality, turn out to be true only 

 in one particular case. In allusion to this circumstance La- 

 croix remarks : u Les series qui expriment sin x et cos x par 

 Tare, sont plus generates que leurs inverses, qui expriment 

 Tare par le sinus ou le cosinus : les dernieres ne conduisent 

 qu'au plus petit des arcs qui ont le meme sinus ou le meme 

 cosinus, tandis que les premieres donnent le sinus ou le co- 

 sinus, quel que soit celui de ces arcs qu'on prenne pour x" — 

 Lacroix, Calcul Diffi et I?it. 9 torn. iii. p. 620. 



Now it is the object of this short paper to show that the 

 series for sin"" 1 ,r, cos -1 ^, &c, when properly investigated, 

 possess the same generality as those for sin x 9 cos x, and that 

 the defective forms above arise from an oversight committed 

 in the analytical processes whence they are deduced. 



If we turn to Lacroix, or to any other writer on the Cal- 

 culus, we find the investigation of the development of y = 

 sin _1 .r conducted as follows : 

 y = sin"" 1 x 



g = (!-**)-* +3** (-**)-* 

 &c. &c. 



The values of these expressions for x = are said to be 



i. dy „ tPy > <Py 



and hence, by Maclaurin's theorem, the first of the above 

 developments is inferred. Now when x = it is not neces- 

 sarily true that y = 0; it is true only when the proposed arc 

 is less than a quadrant. If the arc be greater than a qua- 

 drant, and terminate in the second quadrant, then the value 

 of it for x = can obviously be no other than y = v; if it 

 terminate in the fourth quadrant, its value for x = must be 

 y = 2 it ; if in the fifth, y = 3tt, and so on*. It must also be 

 observed that when the arc terminates in the second quadrant, 



* We are here considering only the positive arcs; the negative arcs 

 having the same sign will terminate in the 3rd, 4th, 7th, &c, quadrants, 

 going round the circle in the opposite direction ; and the corresponding 

 values of y> for x = 0, will obviously be — tt, — 2 ?r, — 3tt, &c, the sam« 

 as before, but with opposite signs. 



