118 



Treatment of Geometry as a branch of Analysis. [No. 134- 



obtuse angle C. We are now in a position to consider (a) and (/3) 

 universal. 



A little further study of the angular functions will contribute to 

 subsequent condensation. In the triangle right angled at C, we have 

 a = c sin A and b = c cos A, dividing one by the other 

 a sin A 

 b 



= a function of A ; specify it as tan A 



cos A 



..a = b tan A. Now b remaining the same, by inspection a will in- 

 crease as A increases, therefore tan A increases with A. This will make 

 sin A increase with A. For cos A (being sin B) is related to B as sin A 

 to A. When A increases, B diminishes ; if then sin A did not increase, 

 cos A would not decrease, and tan A their quotient would not increase. 

 This as relates to acute angles ; with regard to obtuse ones tt — A de- 

 creases as A increases, hence the sin will decrease positively, and the 

 cos increase negatively, the tan of course increasing negatively. 





Table 



of change with angle increasing. 





Angle. 



Sin. 



Cos. 



Tan. 



acute 

 right 

 obtuse 





-|- increase. 



1 

 + decrease. 



+ decrease. 





 — increase. 





+ increase. 

 — decrease. 



It will follow therefore as the sin is a function increasing continuously 

 from to 1, and then decreasing continuously from 1 to 0, as the angle 

 increases continuously from to 7r, that any given value of sin will be 

 found in two parts of this course on either side of the maximum 1 and 

 thus belongs to two angles A and tt — A ; whence there is an ambiguity in 

 determining the angle from the sin, unless there is something to tell us 

 whether it is obtuse or acute. Also if sin B be less than sin A, B 

 may be an angle less than the angle A ; but if A be an acute angle, 

 B may also be an angle greater than the obtuse tr — A. The latter 

 case, however, can never occur when B and A belong to the same 

 triangle, since B + A are always ^L ir and .-. B ^L ir — A. In a 

 triangle therefore if sin A 7 sin B ; A 7 B, and vice versa. 



With the cos there is no ambiguity, the sign -f- or — immediately 

 determines whether the angle is acute or obtuse. If we have 

 cos A = cos B, A = B ; if cos A 7 cos B, A ^1 B. 



