RATIOS OF ANGLES.] 



MATHEMATICS. TRIGONOMETRY. 



615 



PN 



'AP 



Sin. B A P 



Cos. B A P = - 

 A. IT 



Now as A N is measured along A B to the right, A N 

 is positive. And if we reckon lines measured upward, 

 from A towards C positive, it is plain that P N, being 

 measured parallel to that direction, is positive. 



The signs of the sine and cosine of an angle less than 

 ninety degrees are then, by this way of reckoning the 

 signs of the measurements, positive, as they should be. 



Now if we consider an angle B A P., it is clear that P, 

 N' stands in the same relation to B A P , that P N does to 

 BAP. 



Hence we define 



Sin. BAP, 

 Cos. B A P, = 



. AP i 

 It is plain that P 1 N' is positive, and A N' is negative. 



Hence 



The sine of an angle 7 90 ^ 180 is positive, and the 

 cosine of an angle 7 90 ] 80 is negative. 



In like manner, if B A P 2 be the angle subtended by 

 the circumference B P, P 2 , 



P S N' 



Xow P 2 N' is negative, and A N' is negative. Hence 

 the sine of an angle 7 180 ^ 270 is negative, and the 

 cosine of an angle ~7 180 /_ 270 is negative. In like 

 manner if P. A H signify the angle subtended by the cir- 

 cumference 1! P, P., P 3 , 



t. ., 7 270 but ^ 360. 





Cos.BAP, 



And P 3 N is negative, and A N is positive. Hence, 

 ine of an angle 7270^ 360 i negative : and the cosine 

 of an angle 7270^ 360 is positive. These four angles 

 which we have considered are said to be in the first, 

 second, third, and fourth quadrants respectively. 



By means of the above, if we have given the signs both 

 of sine and cosine of an angle, we can tell in what quad- 

 rant it must lie. 



Thus if sine 9 = + m and cos. = n, 9 must lie in 

 the second quadrant ; i.e., must be greater than 90 and 

 less than 180. 



13. To express the Trigonometrical Ratios of any angle 

 in terms of those of an angle less than 90. 



Again, the trigonometrical ratios of any angle can be 

 expressed by means of the ratios of an angle less than 90. 



For if, in the same figure, BAP, B'AP,, B'A P 2 , 

 B A P 3 , are equal to one another, and therefore the lines 

 P N, P, N', P 2 N', P. N, are equal in magnitude ; as 

 also are A N and A N. 



If then we take account both of sign and magnitude, 



P K' P M 



Sin. BAP, = V^- = ^4=in. BAP. 



AP, AP" 



sin. (180 A) = sin. A. 



Sin. BAP., 



PN 



AP ~ AP 

 .'. sin. (180* + A) =-sin. A. 



-sin. BAP. 



Sin. B AP, 



>-sin. BAP. 



Sin. (360- A) = -sin. A. 

 In the same manner it is easy to show that 

 Cos. (180 -A) = -cos. A. 

 Cos. (180+A)=-cos. A. 

 Cos. (360^- A) = cos. A. 



For example, we have seen that cos. 60 = J. 



Cos. 120 s = cos. (180 - 60) =. - cos. 60 = - $. 

 Similarly 



Sin. 315" = sin. (360 - 45) - sin. 45 = - J. 

 If we consider the case of the tangent of an angle, A 

 being an angle, <90. 



sin. (180 -A) -fsin. A 

 Tan. (180'- A) = ._ - ^ A . 



= tan. A. 



cos. (180 + A) ~ -cos. A 

 sin. (360 -A) - sin. A 

 Tan. (360 -A) = cog / 360 _ A ^= .=-tan. A. 



In the same manner we may expr.ess the other trigo- 

 nometrical ratios of angles greater than 90 by means of 

 those angles less than 90. 



It is to be observed, that if we suppose A P to make 

 one complete revolution from AP, it returns to its 

 present position. So that A N and N P are the same 

 both in magnitude and direction for the angle 360 + A 

 as for A. 



The same is true of any number of complete revolutions. 



Hence, if /denote any trigonometrical ratio whatever, 



/(n360 + A)=/(A), 

 where n is any positive integer whatever. Thus, 



Sin. (n 360 -f A) = sin. (A) 



Cos. (i 360 + A) = cos. (A) 

 And so on. 



There are a great variety of relation similar to those 

 above deduced. The following are worth notice : 

 We have before stated that if B A P= A, then P 3 AB 



^ = ~~ -A. 



.'. sin. (- A) = si 



PN 

 ~ AP 



.'. sin. ( A) = sin. A. 

 Similarly, 



Cos. ( A) = cos. A, 

 and .'. tan. (- A) = tan. A. 

 and cot. ( A) = cot. A. 



This result can be arrived at by reference to formulas 

 previously proved. 

 We have seen that under all circumstances 



sin. A. 



.'. sin. (360 -A) = sin. (-A). 

 But we have also seen that 



sin. (360- A) sin. A ; 



.'. sin. (-A) = - sin. A 



14. On the Magnitudes of the Trigonometrical Functions 



of Angles, 0, 90, 180, 270. . 

 The definition of sine BAP tells us that 



^-g. 



Now if A P coincide with A B, A=0 and P N=0 

 /. sin. 0=0. 



If A P revolve round A, P N increases until A P coin- 

 cides with AC, when A=90, and PN=A P. 



.'. sin. 90= 1. 



After which, as A P revolves towards A B', P N 

 decreases until at A B coincides with A B 1 , when P N 

 =0, A=180, and therefore 



Sin. 180=0. 



As A P revolves from A B' towards A C, P N increases 

 negatively until A P coincides with AC, when A =270 

 andPN=-AP. 



.'. sin. 270= -1. 



As A P revolves from A C' towards its original position 

 AB, P N decreases negatively, until when AP coincides 

 with A B, we have A = 360, and P N vanishes. 



.'.sin.360=0. 

 In like manner, if we take the cosine we have 



Cos.A=AT 



Now when A=0 A N= A P .'. cos. =1. 

 A=90 AN=0 .'. cos. 90 =0. 

 A= 180 A N= A P cos. 180= 1. 

 A=270AN = cos. 270= 0. 



A=360 AN=AP cos. 360=1. 



