76 



THE POPULAR EDUCATOR. 



The above results can be verified by constructing a right- 

 angled triangle, as in Fig. 3, with angle A = angle B (. ' . of 45 

 each), where side a = side b, and consequently tan. A = tan. 



45 H _ &= 1, and so on. 



Again, draw A B D, an equilateral triangle 

 (Fig. 5), with the perpendicular B C. 



Then A = 60 and A B C = 30. Also A C = 

 A D = i A B. 



cos. A = = J, .*. cos. 60= = 0'5. 



Fig. 5. 



By (16), sin. 60 = VI ~ cos. 2 60 = </ 1 -^ 



= ,/ = = 0-86602. 

 4 2 



-v/3 

 2 



By (11), tan. 60 = 5^-Jjj! = JL= V3 = T73205. 



Similarly, by (12), cot. 60 = 1-73205 = ' 57735 - 



1 

 By (14), BCC. 60 =~ := 2- 



2 



By (15), cosec. 60 = = 



90c 



As we know the ratios of 60, we of course know the ratios 

 of 30, its complement. 



VI. Supplemental Angles. The supplement of an angle (less 

 than two right angles) is the angle wanting to complete it to 

 two right angles, or 180. Thus the supplement of 30 = 180 

 - 30 = 150; supplement of 175 = 180 175 = 5, and so 

 on. In sexagesimal measure, supplement of A = 180 A. In 

 circular measure, supplement of A = ir A. 



"VTI. Trigonometrical Conception of an Angle Functions of 

 Angles exceeding 90 Use of the Signs -f- and . The trigono- 

 metrical idea of an angle being a quantity to bo calculated 

 rather than, as in Geometry, a shape to be drawn, we find our- 

 selves quite untrammelled by compass and pencil, and may 

 therefore deal not only with angles exceeding 180 which a 

 geometer could only describe as angles turned inside out but 

 with angles of any number of degrees whatever, even exceeding 

 360. We shall, however, find that the functions of every 

 angle exceeding 90 are the functions of some angle below 90, 

 BO that practically we have no need to calculate ratios for angles 

 out of the first quadrant. Indeed, it is obvious that Fig. 2 



cannot possibly be 

 constructed for any 

 angle not less than 

 a right angle. 



It is a conventional 

 arrangement in this 

 science that all posi- 

 tive angles (for de- 

 finition of negative 

 angles see Sect. IX.) 

 are supposed to start 

 from above a kind of 

 horizontal base-line, 

 which forms one side 

 of the angle, the other 

 being supposed free to 

 revolve, in the direc- 

 tion of the arrows in 

 Fig. 6, through an arc of any number of degrees, whether greater 

 than an entire revolution or not. In Fig. 6 let A c be the " base- 

 line" of the angle c A B (less than 90, or "in the first quadrant"). 

 Produce c A to G. Now let A B, the " free side," revolve to the 

 position A D, making D A G = c A B, and A D = A B. Then CAD 

 is more than 90 and less than 180, or is " in the second quad- 

 rant." Now there is clearly no way of constructing, for the 

 angle CAD, the right-angled triangle which played so important 

 a part in Fig. 2, in determining the ratios of the angle then 



180 ; 



Fig. 6. 



being examined, but by dropping the perpendicular D a on to 



CA produced, 

 sin. DAG; 



Sin. c A D is therefore 



.'. sin. c A D = sin. DAG. 



But since DAG = CAB, and triangle A D G evidently = tri- 

 angle ABC, = ?-^- ; 



AD AB 



.'. sin. CAD (an angle in second quadrant) = sin. CAB (an angle 

 in first quadrant). 



But since c A B = D A G, c A B is the supplement of CAD; 

 therefore, generally, 



sin. (TT - A) = sin. A ; > 

 or, sin. (180 - A) = sin. A. j (M) 



From this it appears that the same ratio applies to more than 

 one angle. A remedy for the confusion which might thus arise 

 is found in the following arbitrary use of the signs + and . 



A perpendicular drawn wpward from a given base is con- 

 sidered opposite in sign from a perpendicular drawn downward ; 

 and a line drawn to the rigtit of a given point of opposite sign 

 to a line drawn towards the left from the same point. Con- 

 ventionally, lines measured to the right of a given point are 

 regarded as +, therefore corresponding lines to the left are , 

 and lines drawn upward are +, and downward - . 



By this arrangement it appears that, in Fig. 6, B c, D G, and 

 A c are positive, while c F, G E, and A G are negative quantities. 

 As no negative quantities enter into the ratios of any angle in 

 the first quadrant, its functions are all + or positive. 



We now return to the angle CAD, in the second quadrant, 



and find that its sine also (being, as already shown, P G ) con- 



AD 



tains no negative quantity, and is therefore positive. Formula 

 (28) is therefore correct as regards sign as well as magnitude. 



On the other hand, cos. CAD =_.._ . AG being a negative 

 quantity, we may write cos. c A D = ?. But ^*- = A5 = 



AD AD AB 



COS. CAB, .'. COS. CAD COS. CAB. 



.'. COS. (TT - A) = COS. A ; 

 or, cos. (180 A) = - cos. A ; 



(29) 



And the cosine of an angle in second quadrant is negative. 



Let A D now revolve to the position A E, giving us the trigono- 

 metrical angle CAE, in the third quadrant i.e., of more than 

 180", and less than 270. (This must not be mistaken for the 

 geometrical angle lying below the lines c A, A E, but is the 

 trigonometrical angle subtended by the arc C D E.) Making 

 E A G = C A B, and noting that the lines A G and E G are both 

 negative, but equal in magnitude to A c and B c respectively, it 

 appears that 



sin. c A E = 



E G 



A E 



= sin. CAB. 



COS. C A E = -.-.?-= - G = - ~J* = COS. CAB. 

 A E A E A B 



.-. sin. (180 + A) = - sin. A ; 

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

 and the sine and cosine of an angle in the third quadrant are 

 both negative. 



If A E revolve further to A F in the fourth quadrant, making 

 a (trigonometrical) angle c A F of more than 270, but less than 

 360, then, making c A F = c A B, and noting that E c is nega> 

 tive and A c positive, we find by precisely similar reasoning that 

 sin. (360 A) = sin. A; ) 

 cos. (360 - A) = cos. A. J " 



Thus the sine of an angle in the fourth quadrant is negative, 

 and the cosine positive. 



Generally, therefore (omitting reference to sign), the function 

 of an angle in the second quadrant is the function of its defect 

 from two right angles ; in the third quadrant, the function of 

 its excess over two right angles ; in the fourth quadrant, the 

 function of its defect from two right angles. And since the 

 further revolution of A F into the fifth or any succeeding quad- 

 rant will only involve a repetition of the calculations already 

 gono into, we may still further generalise this statement., and 



