H.ANK TRIGONOMETRY. 



symbol the Greek letter w (pronounced pi). As diameter = 

 radius, we hare 



= 2 w r (2) 



Whence the arc subtending a right angle - -, (3) 



I 



since a right angle is subtended by a quadrant, or one-fourth of 



lumference. 



Let any angle of A be subtended by an arc, a ; then, by the 

 last formula, and by Euclid VI. 33, before quoted 



A : 90 : : o : * r . Whence A = = a. 2 . 



2 90 *_r *r 



2 



Multiplying by 90, A = J< - .- (4) 



From this cither arc, radius, or angle (in common measure) 

 may be found when the other two are given. Thus : To what 

 radius is an arc of 10 feet drawn which subtends an angle 

 of 12? 



Whencel2r= 18 X 10, and r = l-l? ! = 4774 ft. 

 3 14159 3-14159 x 12 



To express the circular unit in sexagesimal measure : 

 By (4), since in this case a = r, 



Circular unit = X -= 



180 

 3-14159 



= 57-29578 (= 206,265"). 



Substituting -%- [see (3)] for a in (1), we get 



Circular measure of right angle = -^. 



p 



II. "Functions" ofanAngle. Although circular measure gives 

 us one means of describing or measuring an angle by lines only, 

 there are other more convenient lines pertaining to every angle 

 than the arc and radius above referred to. They are found by 

 constructing (according to directions given hereafter) a certain 

 simple geometrical figure, the chief parts of which are the angle 

 (which we will call A) and a circle. The lines so produced bear 

 varying ratios to each other as the angle A varies in size ; con- 

 sequently their ratios form measures of the angle. These lines 

 or, more properly, their ratios to the radius to which the 

 circle is drawn are called "functions of the angle," and their 

 ratios to the radius, for any given angle, are always the same, 

 whatever bo the length of the radius. 



The practical utility of this system of lines or " functions " 

 lies in the fact that the figure includes a right-angled triangle, 

 of which the angle A forms part, and that all the functional 

 lines before mentioned either are or may be represented by 

 sides of this triangle. The scale to which the figure is drawn 

 (dependent on the radius adopted for the circle) does not alter 

 the shape of the triangle, or, consequently, the angle-measuring 

 ratios (as we may style them) which exist between its sides. 

 In short, wo have now the means of describing (or measuring) 

 every angle which forms part of a right-angled triangle in terms 

 of the sides, an enormous practical convenience, upon which the 



whole science of Trigonometry 

 .* is based ; for it must be re- 

 membered that all plane recti- 

 lineal figures which require to 

 bo calculated may be split up 

 into such triangles, and thus 

 dealt with in detail. 



To explain the foregoing : 

 Let the angle be D A B in 

 Fig. 2, of less than 90. 

 Placing one limb, AD, in a 

 horizontal position, take any 

 length A D or A B as a radius, 

 and describe the circle D B o. 

 From the extremity of one 



radius, A B, let fall the perpendicular, B c, upon the other. B c 

 is called the sine of the angle B A D to the radius chosen (A B 

 in this case). At the extremity of the radius A D draw the per- 

 pendicular D B, to meet the other radius (produced), n K is 



called the tanyent, and A K the tecant of the angle D A B, to tb 

 radius chosen. 



The difference between an acute angle and a right angle fa 

 called its complement (i.e., the angle lacking to complete or fill 

 up the right angle) ; thus, the complement of DAB in .clearly 

 B A F. A Blight intipection of the figure show* that B a hold* 

 the same relation to B A F that it c holds to D A B ; BO in there- 

 fore the sine of B A r, F H ita tangent, and A H ita secant. Now 

 the function of any angle is Raid to be the co-function of it* 

 complement ; thus B o is the corine, F H the cotangent, and A H 

 the cosecant of DAB, just an the three lines described in the 

 last paragraph are respectively the cosine, cotangent, and co- 

 secant of B A F. It is not, however, usual to speak of " co 

 functions;" all six of the lines described above (or, rather, 

 their ratios to the radius) are called functions of DAB. Two 

 others, not of much utility, are sometimes introduced viz., 

 c D, the versed sine, and the corresponding line o F, the coverted 

 sine of DAB. 



We will now express the above functions of B A c in terms of 

 the sides of the triangle ACS. The functions are the ratios 

 borne by certain lines to the radius in the figure just described ; 

 and as a ratio or proportion may always be expressed in the 

 form of a fraction, the functions may be obtained by dividing 



B C 



these lines by the radius. ' is therefore a correct expression 



A B 



of the value of the sine of B A c, A B being a radius. A B, A D, 

 and A F, being all radii, are equal and interchangeable. So are 

 B o and A c. Moreover, the triangles A F H, EDA, and A c B are 

 evidently equiangular, and therefore, by Euclid VI. 4, the same 

 ratios exist between their corresponding sides ; for instance, 



F H AC 



FH:AF::AC:CB, or = . Bearing these considerations 



A F C B 



in mind, and putting A for the angle BAG, and using the common 

 abbreviations, we get the following list : 



Fig. 2. 



A BC 



sm - A= ' 



cos. A = = . 



A B AB 

 DE BO 



tan. A = = 



AD AC 



FH _ A C 



Cot. A = A F ~ B C 



A E _ AB 



sec. A = A D ~~ A c" 



A H _ AB 



cosec. A= A F B c 



DC _ AD - AC 



Moreover, vers. A = - : 



AB- AC 

 AB 



= 1 - 



And 



** 



AB AB AB AB 



= 1 - COS. A ..................................... (5) 



. FO AF AQ AB-BC . BC 



covers. A = - 



AB AB AB AB 



= 1 Sin. A ........................................... (6) 



Fig. 2 having served its purpose of giving a raison d'etre 

 for this list, and some explanation of the otherwise meaningless 

 names of the functions, may now be laid 

 aside. The right-angled triangle, which is 

 its one claim to notice, reappears in a 

 permanent form in Fig. 3, with its angles 

 indicated by the same capitals as before, 

 and its sides by italics, a being the side 



opposite to A, and so on. C being the right angle, c is the 

 hypothenuse, and 6 is " the side adjacent to the angle." Tte 

 angle B is the complement of A, since the two acute angles in a 

 right-angled triangle must always equal one right angle (for all 

 the angles of every triangle = two right angles). 



To suit the altered lettering, we append a new list of func- 

 tions : 



sin. A = 

 c 



cos. A - . 

 c 



tan. A = - . 

 o 



cot. A = 



sec. A = 



coseo. A = - 

 a 



It is plain that, if we know the numerical value of any one 

 of these ratios, we can find A. In other words, if the ratio 

 between any two sides of a right-angled triangle is given, we 

 can define all the angles. By means which cannot yet be ex- 

 plained, a table of ratios for all angles (in degrees and minutes) 

 under 90 has been drawn up, by reference to which the angle 

 corresponding to any given ratio can be identified at once. 

 This is called the table of natural tines and cosines, and from 



