1896-97.] Prof. Anglin on Circular Functions of 30 and 50. 453 
On the Geometrical Investigation of the Circular Func- 
tions of 3 0 and 5 0. By Prof. A. H. Anglin, 
(Read May 3, 1897. ) 
1. Let the <PAB be 0, where P is a point on the arc of a 
semicircle described on AB of which 0 is the middle point. Then 
making 
< OPC = < OP A, the < PCB = 30 . 
Since the triangles APC, OPC are similar, we have 
AC A APC AP 2 AP 2 
OC “ A OPC “ OP 2 “ AO 2 ' 
AVe shall express each of the ratios sin 30/sin 0, cos 30/cos 0, etc,, 
as a function of AC/OC. 
sin 30 = 
MP MP AP MP AO 
PC AP ' PC AP * OC 
Thus 
sin 3^/sin 0 = ^5 = - 1 
= 4^^ -1 = 4 cos 2 0 - 1 = 3 ~ 4 sin 2 0 . 
CM AM AC 
cos 30 = 
CP CP CP 
Thus 
AM AP AP AM AO 9 AP 
“AP ' CP AO “ AP * OC "AB 
n/1 , „ AO _ AC „ 
cos 30/ cos 0 = - A — - 3 
= 4 ( is ) 2_3=4c0S ^“ 3 
PM _ PM /CM 
tan dtt — — Qp / Qp 
PM AO/OC 
AM ’ AO/OC - 2 ‘ 
