oosnTEs. ETC.] 



MATHEMATICS. TRIGONOMETRY. 



613 



Let x = the number of degrees required. Then x is 

 subtended by an arc of the length of the radius = r. 



Now an angle of 180 is subtended by a semicircle ; 

 i. ., by an arc = ir r, where tr = 314159. 

 . x r = l 

 ' ' 180 ~~ irr ~ ir 



4. The Definitions of the Trigonometrical Lines and 



Ratios. 



It is, however, generally more convenient, and for our 

 present purposes necessary, to determine an angle not 

 by an arc, or by the ratio of an arc to its radius, but by 

 certain straight lines, or by the ratio of certain straight 

 lines to each other. These lines, or, as they are now 

 more commonly regarded, these ratios, are called re- 

 spectively the sine, tangent, secant, cosine, cotangent, 

 or cosecant of the angle. We proceed to define these 

 terms. 



Let A O B be an angle O A. Draw O C perpendicular 



to O A, and 

 with the centre 

 O and any ra- 

 dius O A de- 

 scribe an arc of 

 a circle, meeting 

 O C in C. 



Draw Bn, Bm, 

 perpendicular to 

 OA, to OC; 

 at A and C draw 

 At, Cu, perpen- 

 dicular to O A, C. Then Bn is defined to be the sine 

 of the angle A O B, to the radius O A. 



At Is defined as the tangent of A O B. 

 Ot . . . . secant of AOB. 

 An . . versed sine of AOB. 

 Hence, also Bm (or On) is the sine of B O C. 

 Cu . . tangent of BOO. 

 Ou . . secant of BOC. 

 Now B O C is 90" - A. 



And Bm, Cu, Ou, are denned as being the cosine, 

 cotangent, cosecant respectively of A O B. 

 Hence 



Bn = sine A 

 Bm or On = cosine A 

 At = tan. A 

 Cu = cotan. A 

 Oi = sec. A 

 Ou = cosec. A 



to the radius A. 



To this method there is the obvious objection that the 

 nines, Ac., of a given angle have different values, accord- 

 ing as they are referred to different radii ; accordingly, 

 instead of denning the sines, &c. , as lines, it is, as above 

 stated, now more usual to define them as ratios ; by 

 which means all consideration of the radius to which 

 the sines are referred is avoided. According to this 

 method the definitions are given as follow : 



CAB any right-angled triangle having the right angle 



Then 



BC 



The sine of A is -.- 

 AxS 



"* 



The tangent of A is 



The secant of A is 



Hence The sine of B is 



The tangent of B is 



The secant of B is 



. AB 



BC' 

 AB 



But because A = 90 B 

 The sine of B is the cosine A. 

 The tangent of B, cotangent of A. 

 The secant of A is the cosecant of A. 



BC 

 Hence Sine A = TTT 



BC 



Tan - A =AO 

 AB 



AC 

 Cosine A = -r-g 



AC 

 Cotan. A = ^,=i 



Sec. A = 



Cose. A = 



AB 



AC ISV - " = BC 



N.B. The angle which with another makes up 90 is 

 called the complement of that angle. Hence B is the 

 complement of A ; and the cosine, cotangent, and 

 cosecant of an angle are evidently the sine, tangent, and 

 secant of its complement. 



It is plain (Euc. VI. 4) that the values of these ratios 

 depend solely on the angle, and are quite independent 

 of the magnitude of the sides of the triangles. If, then, 

 we can by any means calculate the value of these ratios, 

 which correspond to any angle, these values can be 

 arranged in a table ; and it is plain that, having such 

 tables, if we have given any one of the ratios defined 

 above, we know the angle ; and vice versJ, if we have 

 the angle given, we know the ratio. 



Such tables have been calculated on principles to be 

 hereafter explained ; for our present purpose it is suf- 

 ficient for us distinctly to understand, that if we have 

 given to us the numerical value of any one of the ratios, 

 we know the angle that corresponds to it, and vice versd. 

 5. On the relations existing between the Trigonometric 



Ratios of the same Angle. 

 Let A B C be a right-angled triangle, 

 A any given angle. 

 C the right angle. 



Then 



Again 



_ _i_ 

 AB 



BO 



=1. 



AB 

 . cos. A + sin. "A = 1 



A B0 

 tan.A=AO 



. . AC 

 cotan. A = ^57=;- 



(Eucl. 47, Book I.) 



(1). 



Again, 



.'. tan. A, cotan. B A = 1 (2). 



tan. A = 



BC 

 AC' 



BC 

 AB 



'AC 



AB 



Again, 



sin. A 

 tan - A = ios.-A" 



. . AB 

 secant A = -j-fj ' 



(3). 



1 



AC 

 AB 



.*. sec. A ' 



cos. A 



(4)- 



In the same manner it may be easily proved that 



cos. A 

 Cotan. A = -- r- 

 sin. A 



Cosec. A : 

 Sec. A 



sin. A' 



tan. A. . . . . 



(6). 



Cosec. A 



It is of very great importance that the student be 

 familiar with the relations we have just established. 

 He will therefore do well to perform the following exer- 

 cises : 



Show that 



(1). sin. A= ^1 cos. 2 A. 

 (2). 



tan. A = <J 4-r- L 

 cos. Z A 



(3). 

 (4). 

 (5). 

 (G). 



tan. A cosec. A = sec. A. 



tan . A + cotan. A = 



sin. A. cos. A 



cosec. A sin. A=cos. A cotan. A. 

 1 -f- cos. A_ 1 

 sin. *A 1-cos. A" 



