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 1 
 
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 32X 
 
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 3 
 
 4 
 
 5 
 
 6 
 
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 Phofesso 
 
 ^1 
 
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 AXALVTIG .\vn ;1(Ai"HCAL 
 
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 1»LAXE A\H sriiK]lil(vVL 
 
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 Hv 
 
 PnoFESSuli „!■■ Civil. KN.yNKn;iM; IN Tin: CjUKCK (U' OTTAWA. 
 
 O'I'TAW \. 
 
 ^1 
 
 :i^ 
 
 I'iiir. t l.y 111.. 
 
 .^iM.'is of 111.' ciJdj) ;-:iir;i'iij:i;i>. 
 
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Prokes 
 
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 j^ 
 
 ANALYTIC AND PRACTICAL 
 
 PLANE AND SPHERICAL 
 
 TRIGONOMETRY 
 
 [ 
 
 ^ot if^t me of gofffges aub ^Aooh. 
 
 By 
 
 Phofessor of Civil Enoinkkiuno f\ thk Com.egk op Ottawa 
 
 OTTAWA 
 
 1878 
 Printed l>y the 
 Sisters of the GOOD SHEPHERD 
 
 :n\ 
 
 vP 
 
 0, M. 1. 
 
 
 i 
 
«vj 
 
 Entered according to Act of i'arllament of Canada, f i 
 in the year 1877, by L. PhiUbert. Paquln, in the office 
 of the Minister of Agriculture. 
 
 I 
 
 
I'llKFACi:. 
 
 The fo'iljwins trtnti.-io, d'^signi'tl I'oi' ejlli",'^', ami lilgh sdiools, 
 and niovf cspcciiUly for ;-ludontH in Eu,yiiiociiug, is iirepiircd 
 uiwii iv thji'jJi^aly now mdhod — It t'tiiljodios uveiy thing nc- 
 (jessiiiy for a courae of Trigouometricid ntudy Huflicienfly extcnsivi' 
 to ('na))lo tlie studfcnt to comprehend n^adily any applications of 
 Trigouonieliy h" may meet with in the l)!wt modern works on ajie- 
 culative !ind applied niathcniaticH. 
 
 A particular attention haa lieon paid to theanvlytical department 
 of the subject. Chapter IV, especially devoted to tliis ) art, eni- 
 bra<!i's til" iiitfltiniportint trigonometrical forn'ulfi' which are found 
 in extensive treatiste, tog-'ther with seveial othei usctful i'orni- 
 ulib not usually found in works devoted to analytical 'J'rigono- 
 ■ iiietry. The Author keeping in view tlie lo.ical and progressive 
 (hivelojmient of the subject, has endeav oured to arrange a conij)lete 
 serii^s of analytical deductions iiecordin^'o a very mt^t hod leal order, 
 which ho belie\es to be a now one. 
 
 Those who wish to limit their studies to thf relations exibting be- 
 twof n the several parts of a triaugh', in as much as it is required 
 for the practice of ordinary Land Surveying, may leave out Chap- 
 ter IV an<l Chapter V without breaks in continuity in so lar as the 
 object they have in view may be concerned. 
 
 The investig-ttion of Trigonometric series being more easily carried 
 on with the aid of Differential ('alculus than with the ordinary inetJi- 
 ods, the Author has given such of ita principles, in the beiriuning 
 of Chf.pterIX,a8 ate necessary for the intoUigence of his investiga- 
 tions on this part of the subject. 
 
 THE AUTHOE. 
 Collfir/e of Ottawa, JonuHr;/ 1878. 
 
 >^l 
 
 
 .1 .. 
 
 I 
 
 
tM 
 
 V' 
 
 AnOUi.AI 
 
 Ml THAI. 
 
 Pnge 
 
 16 line 
 
 2H " 
 .1 ti 
 
 &i " 
 
 (Ml 
 
 72 
 
 74 
 81* 
 87 
 111 
 112 
 
 111) 
 
 KRRATA, 
 
 20 ornl 21;/o-c h read x 
 18/or positive read ncsfltlvo 
 21 /or negative read powltlve. 
 1 
 
 4/or sin 2A read 
 
 8ln iA 
 
 19 /or cos {A + A) read coa{A i-B) 
 \/or coseo B + cosoc A read ooseo B - coHec A 
 12 for 2 Hill ^sln B read - 28lii H Hin fl 
 16/or tan 2»' read tan ' .8 
 10/or fee* .<1 Bco » J9 f cosco ''A ooaoo »fl rwid Bee ''A sec «fl 
 
 cosec »/l c(»scc -B 
 
 16 for t«n (.<1 iB )read sin (.4 t B) 
 28 before the third sine read i 
 5/or sin - if read sin 'iB 
 bfor»\n ^(A r .8) read «ln i(.4 i C) 
 Wfin- (655) read (255) 
 l.S/or(l»))reud(500] 
 
 the second mpmbers or(5()7)and(508) must bo Interchanged, 
 the second meinborm)!" [511] and [512] must, l)o Interchimgo*!. 
 
 Vaiiiation 
 
 Angulaii 
 
 [NVERni: J 
 
 Relations 
 Triangle 
 
 V 
 
tJ^ 
 
 f 
 
 v 
 
 PAiir { 
 
 PLANE TRIGONOMETRY. 
 
 CHAll^EU I. 
 
 I'aob. 
 
 Anou;.au Funotfons 8 
 
 CHAl'li;]! II. 
 
 Ml'TllAI, HKI.ATIONK DETWEK.N AnoUI.AR FuNCTfONH 13— 2'J 
 
 Formulu! fov sine and coaiuu ];) 
 
 f onnulic for tiingoiit and cotangent 15 
 
 Formulas for wecant and cosecant 17 
 
 Formula! for versed flino and covcrned sine 18 
 
 Table of Fundamental Formulie 21 
 
 CHAPTER III. 
 
 Vauiations OF Angulah Functions 22 40 
 
 Algebraic signs of Angular Functions 23 
 
 Numerical values of Angular Functions 30 . 
 
 P'unctions of Negative Angles 41 
 
 CHAPTER IV. 
 
 Angulah Functions involving more than one VAniAni.E 47—92 
 
 Fundamentid proposition 47 
 
 Angular Functions of 1 ^1 + J^l and {A — /f j 48 
 
 Angular Functions of 2^4 ;,63 
 
 Angular Functions of 3.4 55 
 
 (lenoral Formula; for multiple Anglos 68 
 
 Formula) e.xpressing the sum or the dili'erence of two 
 
 functions 61 
 
 < Formulas expressing the product of functions 71 
 
 Formula; expres.ing the quotient of Functions... . 73 
 
 Deductions 77 
 
 Functions involving three angles 80 
 
 Reduction of Trigonometrical expicssions 88 
 
 CmAFTER V. 
 
 IwERW Akqular Functions 92 — 90 
 
 CHAFTER VI. 
 Relations between Angular Functions and sides of 
 Triangles 96 113 
 
 T 
 
 I. 'I 
 
 Jj^: 
 
 
 lil 
 
 i' 
 
 i ) 
 
( •Haiti:;! vn. 
 
 TiiioosoMKTHio Taiii,i:n 113— I'JJ 
 
 (HAITKK VIII, 
 
 Sum Ti')s ur Tku\oi.i;h 12()-13!» 
 
 ]{i;,'lit-Aii;,'lfil Triaii'^'li'M Ilio 
 
 Obliijuo An^'lixl 'I niuiL^ln.-i IJ') 
 
 Um of'Fuhsiiliiiiv Aii;,'U.'s 131 
 
 Ar'ii of 11 Pliiui! Triiiii^,'ln 132 
 
 M I'lisu rt'incnt ol' 1 )iMiunuiH 135 
 
 CIIAl'TKU IX. 
 
 I):vi;r,oi'MKNT OP Axoiii.Aii Kunctidns into hkiiikh 139— K>!) 
 
 (lcupri\l {tiinciiilo of DitfiTontiiiiion 140 
 
 J)ilfi'icnti;iliou of Aiifjular FiinetioUH Ml 
 
 Dillcionliiition of Inverse A nj^uliir Functions I4(i 
 
 .SuocoHnivH l)illor"ntintion of Angitiiir FiinctiunH...l4y 
 
 'lavljr's Tlmorom and Miicliiurin'H llit'orom ir)0 
 
 J''\[ian.sionof Aiigulai Finn;lionH in liuniHoftlicarc!. ITk' 
 
 I'ApaUHion of iiics in tciiii.sol ^iii, cosand tan lr»r) 
 
 Eult'r'H Foiiiiulii', and IM- Moivic'H Formula irwi 
 
 ]Cxllan:^ion of jjoweis of nin j- and co.s .r l.')f< 
 
 Di^vi'lopnicnt of .iin and (!oh of uiultiplti an;,'!"- i<)l 
 
 Ut!.solulion of sin x and cos j: iulo faotois Itl'^i 
 
 (^oinjiutatiou of - ICiS 
 
 Computation of Tiiyononuiliic 'I'abli'H 1(10 
 
 I'AUT 11 
 
 HI'IIEUICAL TIJIGOXOMKTUV 
 
 Objjsox op Spin;nic'AL Tkioonomkthv 100 
 
 CHAPTER J. 
 
 liELATlONB IIBTWKRN 8IDi;s AND AnOLKS OP yPIIHU ThIANOI.KK. 170 
 
 Particular caao of Kiglit- Angled Spherical Triangle 1 6 1 
 
 Napier's rule l'-4 
 
 Oblique Angled Triangle and lu-rpendiculur 184 
 
 CHAPTEIi 11. 
 
 Solution OP Spiiericai, Trianolbh 186 
 
 Eight- Angled Spliorical Triangles 180 
 
 Quadrantal Triangles 1U4 
 
 Oblinue- Angled Spheiical Triangles 194 
 
 k! 
 
 r\ r 
 
PART I. 
 
 PLANK TRKiONOMETRY. 
 
 UlUKCT or TltlOuNuMtTHY. 
 
 TRiauNuiiiCTRY, in its ori({initi and restricted senne, teacheH Low 
 tu tnvSHtigate the relatiuna nubaiHtm^ between the angles and the 
 sides of triangied. 
 
 Plane Trigcnoiuetry treats of pliiue triangles; Spherical Trigono- 
 metry treats of spherical triangles. 
 
 In modern analysis, Trigonometry has a much wider meaning: 
 its object havin'.^ beim extended to all algebraic investigations re- 
 lating to plane angles whether they belong to triangles or not, and 
 its forniuhc, being extensively employed as instruments of calcu- 
 lation in almost every department cf scientific investigation. 
 
 Plane Trigonometry is called PraetinU when rcstricteil to the 
 computiition of triangles, and /lHM///^(V,'rt7 when taken in its cnlar 
 ^ed signification, 
 
 Analiticai. TniooNOMETKV may 1h! defined that branch of Mathe- 
 matics whosi; objret is to investi<iati' fhe vatvre and propcrtieg 0} 
 unijulur functions, am! the ri'liitifittn existimj between them. 
 
 
 CHAITEK 1. 
 
 A Function implies .such a connectiou between two quantities 
 liiat one of them undergoes a change of value when the other, upon 
 which the former is di.'pi-ndent, passes from one state of magnitude 
 tu another. 
 
 4. I 
 
 :■%]■• 
 
 1^ 
 
 IP 
 
I 
 
 1 ■ I 
 
 — 8— 
 
 Anfftilar Functiom are magnitudes which depend upon the an- 
 gular magnitude, tliat is: upon the variable angle that may be con- 
 ceived to extend in the whole angular apace. 
 
 Amjular Spncn is the space which may be run through by a 
 straight line i-evolving entirely about a fixed point in a phiiio .The 
 whole angular space si divided into 360 equal palts called dajreet: 
 «ach degree is divided into 60 equal parts called mivHf.es: G^ah. min- 
 ute is divided iuto 60 e jual parts called seconds. Degrees, uiinutes 
 and seconds are designated by the following marks: " ' " thus 2.}° 
 40'45" is read 25 degrees, 40 minutes, 45 seconds. And Imver sub- 
 divisii'us of a degree are usually expres.sed iis decimals of a second. 
 
 Two straight lines drawn at right angles, divide the whole angu- 
 lar space into four equd parts called quadrants. It is obirious that 
 a quadrant contrins 90''; therefore, a right angle ~ 10'. 
 
 If with a radius of any length, generally taken us unity, a circle bo 
 described about the point of intersection of these two lines as centre, 
 the circumference of this circle will be 
 the linear representative of the whole 
 angular space about that point. And 
 •since^the angles at the centre of a cir- 
 cle are proportional to the arcs of the 
 circumference intercepted between their 
 sides, and are measured by them, the an- 
 gular functions may, also, be concei\ cd 
 as dependent upon these arcs: u n d e r 
 Hhis respect, they are called c/rrwiar func^Vws and, also, ^•(V/')/(0- 
 metricallmest which will be defined hereafter. 
 
 The complemen: of an angle or arc is the difference between 90' 
 and that angle or arc. 
 
 The supplement of an angle or arc is the difference between 180° 
 and that angle or arc. * 
 
 V'e may take the following method of illustrating the generation 
 of the angular magnitude, 
 
— 9— 
 
 
 art'Dce between 180° 
 
 ftting the generation 
 
 Let angles bs conceived as generat- 
 ed by the motion of a straight line re- 
 volving from a fixed line CA, about 
 a fixed point C: when the revolving 
 line coincides with OA, there is no an- 
 gle formed, or the angle = 
 
 When it assumes successively the po- 
 sitions of CP,CB, CF, CA', CP", CB' 
 CP'", the angle generated, are respective- 
 ly an acute angle, a right angle, an obtuse angle less than two right 
 angles, an angle equal to two right angles, an angle greatei than two 
 and less than three right angles, an angle equal to three right angles, 
 an angle greater than three and less than four right angles. 
 
 When, finally, the generating line has completed an entire revo- 
 lution, it will coincide again with C/1, having formed an angle e- 
 (lual to four right angles. 
 
 If we suppose the revolutiop to recouunence, then the genera- 
 ting line will form angles greater than four, than five, or any other 
 number of right angles. 
 
 Now any point A on the moving line, will, at the same time, de- 
 scribe the circumference ABA'Ji' about C, and may be consider- 
 ed as the generator of all arcs that may be taken on the circumference 
 
 Let A be the origin of any arc, and the arcs to be generated by the 
 motion of a point in the direction upwards, then A A' will be call- 
 ed the diameter passing through the origin, B the o vigin of the com- 
 plements, since the complement of an <irc is the remainder obtained 
 bysubtraetiug that arc from a quadrant or 90°,55' the diameter pass- 
 ing through that origin; — ^' the origin of the supplements, and 
 A A' the diameter passing through the origin of the supplements. 
 
 Wherei the generating point is conceived to stop in its motion,there 
 is the end of the «to generated. Thus the generating point being at 
 jP, the afc geileiMed is A P, its complement 5P, its supplement A 'P 
 
 The geneMting point being at P', the arc generated is AP', ita 
 .complem^lit 3P', its atipplement A'P'. 
 
 The generating point being at P", the arc generated is JP", its 
 complement BP", its supplement i4'P". 
 
—10— 
 
 Iho generating point being at P'" the arc generated is AP'" 
 its complement BP'", its supplement A'P'". 
 
 Hence it is seen that the points P,P',P",P"',aiTe respectively the 
 ends of the arcs AP,AP'^AP"^AP'",a3id of their complements and 
 their supplemental that is the end of an arc,that of its complement 
 and of its supplement are the san. e point. 
 
 It is obvious also that the complement of an arc less than 90'' is 
 po8itive,and that the complement of an arc greater than 90' is nega- 
 tive, because by the rule of algebraic subtraction, we have 
 90— [90— a] = b 
 90 — [90 + a] = — <- 
 
 For the same reason the supplement of an arc less than 180' is 
 positive and that of an arc greater than 180' is negative, that is: 
 180— [180 — a] = h 
 180 — [180 -L- a] =z—h 
 
 The Sine of an arc is the perpendicular let fall fiom the end of 
 the arc to the diameter passing through its origin. 
 
 The Tangknt of an ire is the perpendicular erected at the origin 
 of the arc on the diametet passing though the origin and termi. at- 
 ed by its intersection with a line drawn from the centre 'hrough 
 the end of tlie arc. 
 
 The Secant of an arc is the line drawn from the centre through 
 the end of the arc, and terminatiul by its intersection with the h\i\- 
 gent. 
 
 The Versed Sine of an arc is tluit part of the diameter passing 
 through the origin, intercepted between the origin and the foot of 
 the sine. 
 
 Thus for tho sake of illustra- 
 tion we have in the annexed fig- 
 ure; 
 
 PM = sine of the arc AP 
 
 AE = tanyent " " 
 
 OE = secant " " 
 
 AM = versed nine " " 
 The same definitions being ap- 
 plied to tho trigonometric lines of 
 

 \ 
 
 -11— 
 
 rated ia AP' 
 
 t'loiu the end of 
 
 the arc BP^ complement of AP^ and whose origin ia at B we have 
 
 PL=8ine of the arc BP 
 
 BR= tangent " " 
 
 CR=8ecant " " 
 
 BL— versed sine " " 
 But because the arc BP is the complement of ^P, we have: 
 
 PL=sine of the complemen' of AP 
 
 BIi=tan<jent " " 
 
 01i=secant " << 
 
 BL= versed sine " " 
 
 And for abreviation we write, by a contraction of the word com- 
 plement: 
 
 cosine for sine of complement 
 
 cotangent for tangent of complement 
 
 cosecant for secant of complement 
 
 coversed sine for vcfsed sine of complement 
 And for further abreviation, '^e write 
 
 sin for sine 
 
 cos for cosine 
 
 tan for tangent 
 
 cot for c; tangent 
 
 sec for secant 
 
 cosec for cosecant 
 
 covers for co versed sine 
 Becatiso the sine of an arc is always parallel to the diameter BB' 
 passing through the origin of the complements, it follows that the 
 cosine PL is e^iual to that part of .he diameter ^^'intercepted be- 
 tween the jontro and the foot of the sine; and therefore in practice 
 the cosine is taken as being CM, and is defined the distance be- 
 tween the centre and tlie foot of the sine. 
 
 The trigonometric lines of an arc, or Circular functions, depend 
 on the magnitude of the radius of the circle; for the greater is the 
 radius of a circle the greater will be the arc included between the 
 sides of an angle at the centre, and the greater also will be the cir 
 cular functions relating to that arc, as is evident by the inspection 
 of the annexed figure. 
 
 I 
 
 
■ 12- 
 
 ''1 
 
 But from tlie simil- 
 itude of triangles, 
 we have 
 
 Cp " ' 'P 
 ae AE 
 
 Ua~ (Ia 
 etc. 
 tliat is, the ratios of 
 ciicui--r funciions 
 of an arc of a gieat- 
 flr circle to tiie radi- 
 us of that circle are 
 e(iual to the ratios 
 of circular functions of an arc of a smaller circle to ita radius. 
 
 These ratios constitute properly the Angular Functions of the 
 angles subtended l)y the arcs, !^o that the sine of an angle is the ra 
 tio of the sine of the included arc to the radius of the circle; and in 
 general the atKjulor functmis are the ratios of the corref] cmUvfj 
 cin'ulnr fun tioim to the radius of the circle. 
 
 It is obvious, from the last definition, that the angular functions 
 are indepentb.'nt of tlvj magnitude of the radius. 
 
 Hut since in practice, the radius of the circle is always assumed 
 as unity, tiien the anguLir functions are equil to the corresponding 
 circular functions; and it must lie l)Orne in mind that "circular func- 
 tions" "anj,'ulav functions" and "trigonometric lines" all designate 
 the same maguitd s. 
 
CHAPTER II 
 
 Mutual Relations between Angulab Functions 
 
 tt'i 
 
 Formulas /or Sine and Cosine 
 
 N. B. In the following investiga- 
 tions, the radius will be assumed 
 
 •adius. 
 
 ^^^^^H 
 
 
 as unity. 
 
 
 'IONS of the 
 
 ^^^^^^1 
 
 
 
 
 ;le is the ra 
 
 ^^^^^^M 
 
 
 
 
 rcle; and in « 
 
 ^^^^HH 
 
 
 Since PAP -\- CAP = 
 
 CP^ 
 
 )rret'] ctuiiny 
 
 ^^H 
 
 
 or sin* + cos* = 
 Therefore sin = ^[1 - 
 
 - cos*] 
 
 ar functions 
 
 i^^i^^i 
 
 
 
 
 
 Since 
 
 CP.i 
 
 PM:-CE: AE 
 
 
 pys iissumed 
 
 ar 
 
 1 : sin : : sec : tan [6] 
 
 
 rresponding 
 ircular func- 
 
 rhijrofoi'c 
 
 tin 
 '^'"^ sec 
 
 [2] 
 
 nU designate 
 
 Since 
 
 CB: PM :: BR: CM 
 
 
 
 ur 
 
 1 ■ sin : : cot : cos ['.'] 
 
 
 
 I'lioiet'oie 
 
 cos 
 
 sin = 
 
 cot 
 
 m 
 
 
 Siucf 
 
 (■A : (\\I: -.AE-.PM 
 
 
 
 ji' 
 
 1 : cos : 
 
 : tan : sin [ti] 
 
 [4] 
 
 \ • I 
 
 Therefore 
 
 sin = tan x cos 
 
 Since 
 or 
 
 Therefore 
 
 CB: PM:: CR :CP 
 
 1 : sin : : cosoc : 1 [e] 
 1 
 
 sm = 
 
 cosuc 
 
 [5] 
 
-14- 
 
 I t. 
 
 By the definition of the trigonometrical lines, we know that the 
 Sine, tiingent and secant of an angle or arc. are equal to the cosine,i 
 cotangent and cosecant of the complement of that angle or arc. Thus 
 denoting by a any assumed arc, we have 
 
 sin (90° — a) = cos a 
 
 tan ( 90° — a ) = cot a 
 
 sec (90° — a) = cosec a 
 These relations give rise to the following deductions: 
 Because sin = ^ [1 — cos*] , ly (1) 
 
 we have sia ( 90° — « ) = ^ [1 — cos' (90° — a)] 
 hence cos < = V [1 ~ sin'' a] 
 
 or generally, cos = ^ [1 _ sin'] (6) 
 
 Deduction verified by the equation [^t] 
 
 Because 
 
 Sim 
 
 tan 
 sec 
 
 , by (2) 
 
 we have sin ( 90° — a ) = tan(90° — (?) 
 
 sec(90° — rt) 
 cot a 
 
 hence 
 
 or generally, 
 
 cos a =, 
 
 cosec u 
 
 COS=_J£L_ 
 
 or 
 
 cosec 
 Deduction verified by the proportion 
 CP:PL = CM ■.-.CR: BR 
 
 1 : cos : : cosec : cot [^] 
 
 (7) 
 
 III 
 
 ; ■ 
 
 Because 
 
 sin =i^, by [3] 
 cot 
 
 hence 
 
 or generally 
 
 cos a = . 
 
 cos =_ 
 
 tan 
 
 we have sin (90 — a)= _££i(^ — flL 
 
 cot (90° — a) 
 
 sin a 
 
 tan a 
 
 sin. 
 
 Deduction verified by the proportion [</] 
 
 (^) 
 
-15— 
 
 know that tho 
 1 to the cosine,! 
 lo or arc. Thus 
 
 Because iin = ten x cos, by (4) 
 
 we have sin (90° — a) = tan '(90° —a) x cos (90° — a) 
 
 hence cos a = cot a x sin a 
 
 OT generally, cos = cot x sin 
 
 Deduction verified by the propoiiion [e] 
 
 Because sin = 
 
 we have ain (90° — a) = 
 
 hence 
 
 or generally, 
 
 1 
 
 cosec 
 
 -by [6] 
 
 1 
 
 cos (I = 
 
 cosec (90° — a) 
 1 
 
 sec a 
 
 ^ 1 
 
 COB = 
 
 sec 
 
 or 
 
 Deduction verified by the proportion 
 GA : CM:: CE: CP 
 1 : cos : : (tec. : 1 
 
 (9) 
 
 (10) 
 
 <F 'I 
 
 r 
 
 ^ 
 
 ii I 
 
 Formulas for tangent and cotangent 
 
 By the Proportion [rf] we have 
 
 tan = 
 
 sin 
 cos 
 
 Since 
 or 
 
 we have 
 
 CA : AE: : BR : GB 
 
 1 : tan : : cot : 1 [j/] 
 
 1 
 
 tan = 
 
 cot. 
 
 By tlie proportion [6] we have 
 
 tan := sin x sec 
 
 Since 
 
 or 
 
 we have 
 
 CE'-AE'-^GA^ . 
 sec* = tan * 4- 1 [A] 
 tan= -x/C^ec* — 1] 
 
 (11) 
 
 (12) 
 (13) 
 
 (14) 
 
 s 
 
 1^1 
 
.1«— 
 
 ^N 
 
 
 Because = 
 
 we have t*n (90° — a) = 
 
 sin 
 
 by (11) 
 
 hence 
 
 or generally, 
 
 cot o = 
 
 cot — • 
 
 CM 
 iilB(90°--a)_ 
 
 cos a 
 
 sin a 
 
 coa 
 
 sin 
 
 , see page 14 
 
 Because 
 we have 
 hence 
 or generally, 
 
 Deduction verified by the proportion [c] 
 1 
 
 tan = 
 
 tan( 90° — a) = 
 
 cot 
 
 cot (90 — a) 
 cot a = -- 
 
 cot = 
 
 tan a 
 
 1 
 tan 
 
 Deduction verified by the proportion [^] 
 
 Because tan = sin x sec, by (13) 
 
 we have tan (90° - a ) = sin (90° — o) x 8ec(90° —a) 
 
 hence cot = cos a -j- cosec a 
 
 or generally, cot = cos -f- cosec 
 
 Deduction verified by the proportion [/] 
 
 Because tan =v' [sec* — 1] 
 
 wo have tan (90° — a) = V [sec* (90— a)— 1] 
 hence cot a =: -^/ [cosec* a — 1] 
 
 or generally, cot = ^/ [cosec* — 1] 
 
 Deduction verified by the equation 
 CIP = £JS? + CB* 
 or cosec* = cot^+l [»] 
 
 (16) 
 
 (18) 
 
 (17) 
 
 (18) 
 
-17- 
 
 Formulas for Secant and cosecant. 
 
 By tho ofiuation [A] we have 
 
 8ec= Vfl'f-L/J] 
 
 By the proportion 1 : cob : : see: 1, see page 15, we have 
 
 1 
 
 800 = 
 
 C08 
 
 By the proportion [i] we have 
 
 tan 
 
 sec = 
 
 sin 
 
 IcMaM aec - V [1 + tan' ], by (W) 
 
 we have sec (90° — a) = ^[l-\. tan»(90^ — a)] 
 hence cosec a = V [1 + cot* a] 
 
 or generally cosec = ^ [1 -j- tot*] 
 
 Deduction verified by the equation [i] 
 
 Because 
 
 2^ 
 
 cos 
 
 seo = by (80) 
 
 1 
 
 we have sec (90° — a) = — 
 
 cos (90° — a) 
 
 1 
 
 cosec a = 
 
 hence 
 
 or generally, 
 
 Because 
 
 cosec = 
 
 sin a 
 \ 
 
 sin 
 
 Deduction verified by the proportion[e] 
 tan 
 
 sec =- 
 
 by (21) 
 k sin •' \ ' 
 
 we have sec (90° — a) - <»P(90°— ct) 
 
 Bin (90" — a) 
 
 cot a 
 
 coMc a = 
 
 cosa 
 
 hence 
 
 (19) 
 
 (20, 
 
 (21) 
 
 (22) 
 
 (23) 
 
 [ 
 
 f 
 
-18- 
 
 cot 
 
 or generally cosec - ^^^ 
 
 Deduction verified l>y the i)roi>ortion[/] 
 
 Fwmuhin fur Vcrned Sine iiiid Con-rsi-U Sim: 
 
 [24] 
 
 I* 
 
 'By deUuitiou(/»»;/e 10) wo bavt- 
 
 VITH = ^1 M 
 
 But AM = ('A — CM 
 
 Thorelbrc vera = 1 — cob 
 
 BecauHe vers = 1 —cos, by (25) 
 
 we have vers 1 90° — a) = 1 — cos (90° —a) 
 
 hence covcis = 1 — sin a 
 
 or generally 
 
 coveic = 1 — sin 
 
 (25) 
 
 (26) 
 
 Beeides tlio ahove formula , many others may he obtained by com 
 bining them. The moat iismlly emjiloyed are the following; 
 Substituting (19) in (2) givi-s 
 
 tan 
 
 sm = 
 
 V[l + tan'] 
 
 Substituting (18) in (3) give; 
 
 cos 
 ^ [cosec* — 1 J 
 
 Substituting (22) in (5) givi-s 
 tin = 
 
 Substituting (22) hi (7) gives 
 cos = - 
 
 Substituting (14) in (8) givea 
 cos = 
 
 1 
 
 V [14- cot' J 
 
 cot 
 
 V LI + c"t*] 
 
 sm 
 VT^ec"'— IJ 
 
 (27) 
 
 (28) 
 
 (29) 
 
 ( 30) 
 
 i31) 
 
—19— 
 
 Substituting (19) in (10) gives 
 
 1 
 
 ''««=V[l+tan'] 
 Substituting (6) in (11) givoa 
 
 tiaa = 
 
 sm 
 
 V[l — sin'] 
 
 Substituting (1) in (11) gives 
 
 cos 
 
 Substituting (1) and (6) in (11) gives 
 
 V[l — 3in»] 
 Substituting (17) in (12) gives 
 
 tan = 
 
 1 
 
 cos X coseo 
 
 Substituting (1) in (16) gives 
 
 cot = 
 
 cos 
 V[l — COS*] 
 
 Substituting (6) in (16) gives 
 
 sm 
 
 Substituting (6) and (1) in (15) gives 
 
 ^ V[ l -8in'] 
 
 Substituting (13) in (16) gives 
 
 cot = 
 
 sin X see 
 
 (32) 
 
 (831 
 
 (34) 
 
 (35) 
 
 I 
 
 (361 
 
 (37) 
 
 (38) 
 
 (39) 
 
 (40) 
 
,'(J— 
 
 1*^^ 
 
 Hiilwliluting (0) in (20) given 
 
 HOC = 
 
 1 
 
 Hiilwtituting iDi in (20) gives 
 
 HOC = 
 
 1 
 
 Bin X cot 
 
 Mutwtituting (1) iu I'il) givcM 
 sec = • 
 
 Subntituting (li in (23) gives 
 COdt'C = - 
 
 SubNtitutiug (4) in (23) gives 
 cosec = 
 
 SubBtituting (6i in i24) gives 
 cosec = 
 
 tan 
 
 V[l-C08»] 
 
 V [1— cos*] 
 
 1 
 
 tan X cos 
 
 cot 
 
 V[l— 8in«] 
 
 (41) 
 
 (431 
 
 (43) 
 
 (44) 
 
 (46) 
 
 (46) 
 
 By the above deductions wo see tlmt Sine and Cosecant are le- 
 ciprocul to each other, oa are, also, Cosine and Secant, and Tangent 
 and Cotangent; ho that we may write 
 
 Sin :< Cosec = 1 
 
 Cos :< Sec = 1 
 
 Tan X tot = 1 
 
 The above results being of the highest importance in all funda- 
 mental investigations, are collected and arranged in the following 
 table, in which u denotes any angle or arc. ^ 
 
 ■■H 
 
-L'l- 
 
 ).4 
 
 ^secant are le- 
 at, and Tangent 
 
 36 in all fnnda- 
 1 the following 
 
 TADLE I. 
 
 Fundamental Koiimulas. 
 
 Hiu « = y'[l — cos* 
 
 A 
 
 Cos 
 
 a 
 
 = ^/ [1 — sin* a 
 
 tan 
 
 
 
 
 cot a 
 co8ec a 
 
 _ COH ll , 
 
 
 
 
 mn u 
 
 cot II 
 
 tan (( 
 
 — CO* (( tan It 
 
 
 
 
 =• pin fi col rt 
 
 1 
 
 cosec a 
 
 
 
 
 _ 1 
 
 ~ sec a 
 
 tan a 
 
 
 
 
 cot 
 
 Tana = 
 
 V [l+t«n* «] 
 
 cos o 
 ^ [cosec' a — 1] 
 1 
 
 V [1 4-'cot» a ] 
 
 sin ti 
 
 cos r/ 
 
 I 
 
 V [l + cot» rt] 
 sin u 
 
 ^[ sec'a — 1] 
 1 
 
 Lot « — — : — 
 
 cot a 
 = sin <i sec « 
 = v^ [sec* « — 1 ] 
 
 sin (( 
 
 ^ [1 — sin* rt] 
 _ v^ [1 — cos* a] 
 
 cos (I 
 
 VJl — cos* rt] 
 "[ r^i n» a] 
 
 1^ 
 
 cos It cosec a 
 
 V [l + t»n*a] 
 
 cos a 
 sin a 
 
 1 
 
 tan f/ 
 
 cos a cosec a 
 
 [cosec* '/ — 1] 
 
 cos a 
 
 ^/ [1 — cos' rt]~ 
 V [1 — sin* a] 
 
 sm a 
 
 = 7- 
 
 [i — sin* rt] 
 [1 — cos* o] 
 
 1 
 
 sin a sec a 
 
 r 
 
 te^ 
 

 B 
 
 Soc « = V [ ^ + ^^^' "] C°^°<^ ''^ = V [1 + cot* «] 
 1 1 
 
 COM ft 
 
 tan a 
 sin a 
 1 
 
 1 
 
 ^[1 — sin* a] 
 1 
 
 8in rt cot a 
 tan a 
 
 ^[1 — cos* a] 
 
 1 
 cos a tan a 
 cot a 
 
 ^[1— co8*«] ^'[l— sin'a] 
 
 Vers a = 1 — cos a Covers ft = 1 — sm ft 
 
 CHAPTER III. 
 
 Variation of Angular Functions. 
 
 By the definition of a function given in Chapter I, [page 8] , it 
 is obvious that the term function implies variability, or that two or 
 more quantities vary together in accordance with some mathemati- 
 ical law. 
 
 Any one of two quantities connected by a functional relation, 
 may be assumed to vary arbitrarily; it is then called the independ- 
 ent variable, whilst the other, which varies dependently of the 
 former, is called the dependent variable. For the latter, the term 
 function is generally reserved 
 
 In angular functions, the independent variable is the angular 
 magnitude; and the trigonometric quantities relative to that angle 
 or to the arc measuring that angle, are the dependent variables. 
 
 Hitherto wo have established certain relations between angular 
 functions independently of their variation in value. We shall next 
 
 ■i 
 
-23— 
 
 prococd to invL'stigato tlie jirinciplcs which rule this variation. 
 
 Angular i'uuctioiiK vary in two ways; 1st in their relntir value, 
 or in respect to algebraic signs; 'Jiid in their (ili.iiiliifr or iinin/'ri- 
 I'ltl value; 
 
 Ahjehrtiii' si(iiis m- .' u ii'iir FnnrVciis. 
 
 The Algebraic .Symbols -f and ai<' not only indicative of the 
 y-/'f/'rt/'W,< of uldition and su1)traction, ms dnHned in the beginning 
 of elementary wo-ks on Algiibra, but aw, also, indicativp of the 
 (jualitiei of (luantities. And, far from coul'usion arising from this 
 double mode of interpreting algebraic syndiols, analysis, on ;hi' 
 contrary g.iins thereby considerably in i)jwer. 
 
 Let (' be a lixed point on a fixed stniight line ,11. Then to 
 .1' C A 
 determine th? pjiiiiou of .inv pjiut; on t'nt liie, we must 
 know the distance of that point from C and, also, on ii/i,r/i 
 tti'ilc of C the point lies. Tln^se two opjiosite diriictions are respect- 
 ively indicated by the signs + and — sothit, if tho distances meas- 
 ured from C towards the right hand be th-noti'd by -f, the distancps 
 towards the left hand will be denoted l)y — ; thus, if A and A are 
 both distant 4 from C, the distance of A s denoteil by -f- 4, and 
 that of .4' by — 4. 
 
 Again, let A'A" ami )' Y bo the two tixed straight lines which 
 meet at right angle at A. 
 
 To determine the position of any \ o.'nt /•, in the ])lane co tp.in- 
 
 ing the two lines, we ought to 
 know the distance of F from each 
 of these lines, and, also, on which 
 side of each it lies. The two ojtpo- 
 site directions respecting each of 
 the lines JTJC ,and )' Y , are dis- 
 tinguished by the signs' -f- and 
 — , as before; so that, if the dis- 
 tances measured from A'A' in the 
 direction ohorr be indicate d bv + 
 
 1' 
 
 H 
 
 \X 
 
 P' 
 
 'F 
 
 M. 
 
 X 
 
 P' 
 
 V 
 
 ,,,,) 
 
 T 
 
 those measured from the same straight liuL in tji ' "lirecjtion bel 
 
 ow 
 
 ■Mvll 
 
 '• ' ■ \ 
 
 4.1 .i 
 
 ■ii' i) 
 ! i 4 
 
 i..n 
 
'f:'" 
 
 if 
 
 ii! 
 
 :;4— 
 
 iiiv iiidiLMtfd by — ; ;m(l tlic distancs moiwsuirii iioiii YY towards 
 llio riglit h;ind ])fiiig walked l)y 4 , those tow.uds tlio left Imnd 
 i.w diisi-nated l)y — . Tliu.-i, if P li.-.s iu tho iiist quiidnuit A AY, 
 liciJi dl^-ll,nc(•s PMand J'li -mv. iio-sitivc; for P' lying in ti o. second 
 iliiadiiint .V^l y we have P jY iiositivo and F'Ji negative; for F'' 
 lying in the third JC'A 1", the both distances F\\ and P"^'areneg 
 ative; an.l for P" lying in tho fourth (luadrant XAY*, we have' 
 F"M negative, and P"'S positive. 
 On this convention rests th<! principle that rules the signs of Trig- 
 ouonietiic lines. 
 
 All lines peqiendicular on the diameter A A', and lying above 
 are considered as positive, or have tho sign -f; and those lying be. 
 low have the sign — . 
 
 All lines perpendicular on the diaiiutor BB' and extending to- 
 l wards the right, are considered as positive, or have the sign-f-°iiud 
 those extending towards the left have the sign — . 
 
 Thus it is seen by the in- 
 8i)ection of the figure th:it 
 for an arc .IP, not exceed- 
 ing the first quadrant or 
 90 , both the sine P, J/, 
 and the cosine P, /-,, or i'.s 
 equal r^^,, are positive. 
 
 In tho second quadrant 
 the sine PJ/., is positive' 
 and the cosine PJ^^, or its 
 equal CMj is negative. 
 
 In tJie third quadrant, 
 both the sine PM^, and 
 the cosine P.,//,, or its e- 
 qual CM:i fire negative. 
 ., In tho fourth^quadrant, 
 tho sine P^Mf is negative; 
 and the cosine P^Lj, or its 
 
—25— 
 
 e signs of Trig- 
 
 il lying above 
 those lying be. 
 
 extending to- 
 he sign -f-; and 
 
 equal CM^, positive, that is, supposing a to donoto any number of 
 degrees less than 90; we have 
 
 -f sin (90° — a) 
 -j- sin (90° + a) 
 — sin (180° 4- a) 
 -sin (270°+ a) 
 
 (47) 
 (48) 
 (49) 
 (50) 
 
 +C08 (90° — a) 
 
 — cos (90° + a) 
 
 — cos (180° + a) 
 + cos (270° + 0) 
 
 (61) 
 (52) 
 (53) 
 (54) 
 
 The signs of the sine and cosine being determined, the signs of 
 all the other Trigonometric lines may be at once established by sub- 
 stituting the above values in (11), (15), (20), and (23), Thus: 
 
 + 8in (90° — a) 
 
 + COS (90^ — 0) 
 
 -=+tan (90°— fl) 
 
 + 8in (90° + a) 
 — COS (90° + a) 
 
 = — tan (90° + a) 
 
 (55) 
 (56) 
 
 fl 
 
 « I 
 
 4 
 
 — sin (180° + a) 
 
 — cos (180" + a) 
 
 = +tan (180° + a) 
 
 JZ^!?JS+^ = -tan(270° + a) 
 + COS (270° + «) ^ 
 
 (67) 
 (68) 
 
 + co8(90°-a) ^^,,t(90°-«) 
 + sin (90° — a) ^ 
 
 -«'M90° + a)^_,,t(90° + a) 
 + 8in (90° + a) ^ 
 
 -cos (180^ + a) ^ _j_ ^^ jgQo ^ «) 
 — sin (180° + «) ^ ^ 
 
 , (69) 
 (60) 
 
 (61) 
 
t!* 
 
 ,: 
 
 *: 
 
 >!• 
 
 is 
 
 fi- 
 
 -fuos (270° -f (J I 
 — sin"(2W'~-f ai 
 
 1_ 
 
 -|-coH |90' — (t) 
 
 1 
 
 1 
 
 — cos (180" +rt) 
 1 
 
 + COS (270° + ai 
 
 1 
 
 -f sin 1 90-'— a) 
 
 1 
 
 + wn i90° + rt 
 
 1 
 — iiiu ,180°+ ,1 
 
 -2G — 
 
 '^rrzl: = — pot i270'' 4- a\ 
 
 = +sec (90' — (i\ 
 
 = — see i90°4-,0 
 
 = — sec ;18(i°-l-(n 
 
 = + see (270' -f<M 
 
 - +cosec 90° — -0 
 
 = -|-co*'>JC '90' + "l 
 
 g , — = — coacc ilHO' -]•- a I 
 
 1 
 
 n^y27T)^-j-<(T 
 
 . co.sec (2iO' -|- r<i 
 
 1 02' 
 
 (G3i 
 
 (64) 
 ,65) 
 (66, 
 
 167) 
 (68) 
 
 Hunce it is soon that Tangent and (.'otangont have tlie same sign 
 in each quadrant; and both are positive wlien Sine and Cosiu(! have 
 alike sign, and negative wlien they have unlike sign. 
 
 Secant has always the saiiie sign as Cosine; anil Cosc^caiit Ims a!- 
 wayb the yaine sign as .Sine. 
 
 Ta 
 
 Co 
 
 And for thi 
 
 Ta 
 
 Cc 
 
 And for th 
 
 T; 
 
 C( 
 
 And for th 
 
\ 1 1.'. 
 
 Versed sine ami (^oveiHwl aino are positive in the four quad- 
 rants becaute^ as seen in (45) and (40, they aie always equal to 
 the radius or unit, diminished respectively by Cosine and sine, 
 which never exceed unity. 
 
 These analytical results may Ije verified by the inspection of the 
 figure below, where we have, for the arc AP, not exceeding the 
 first quadrant, 
 
 i- 
 
 Tangent yl El joo«Yu'e,being above ^ A, 
 Cotangent B/?, positive, being to the right of BB, 
 And for the arc AP^ in the second quadrant: 
 
 Tangent AEt negative, being below AA, 
 Cotangent Blt^ negative^ being to the left of BB, 
 
 And for the arc AAxP^ in the third quadrant: 
 
 Tangent AE, positive,hoing above A A, 
 Cotangent BRi positive, being to the right BB^ 
 
 
 ( 
 
 4 
 
 And for the arc AA,Pf in the fourth quadrant; 
 
11-, 
 
 
 -28- 
 
 W<' 
 
 l 
 
 m. 
 
 it?"; 
 
 Tangent AEj negative, being below .4^4,, 
 Cotangent BB^ negative, being to the left of BBi 
 
 Where we see a perfect agreement with the results (55^, (56) 
 (57) (581, (59), (60), (61), and (62l. 
 
 Seoant and Cosecantare estimated from one and the same point 
 (the centre of the circle,) whatever be the arc they are relating to; 
 therefore, each of them has not its opposite directions referred to 
 the diameters AA^ and BB^ but to the extremity of the arc, 
 so that they are considered positive when estimated towards the 
 end of the arc, and negative whsa Jitimitoi in an opposite direc- 
 tion. Thus we have for the arc AP, in the first quadrant: 
 
 Seoant CB, positive; 
 
 Cosecant CBi positive; ' .- 
 
 And for the arc APj in the second quadrant: .. 
 
 Secant CE^ negative; ', 
 
 Cosecant CBj positive; 
 And for the arc AP^ in the third quadrant: 
 
 Secant CE, positive; • 
 
 Comcaat CR, negative; ' 
 
 And for the arc APj in the fourth quadrant: 
 . Secant CE, negative; 
 
 Cosecant CR, negative; 
 which all agree with (63), |64', i65i, (66i, i67», ',68', (69i, and (70). 
 
 i;^ 
 
 I 
 
 Versed Sine and Coversed Sine beii g always estimati'd in tho 
 fame direction^ the iirat from A towards A^, and tho second from 
 B towards Bi, must not change in signs, and therefore remain pos- 
 itive in the four quadrants. 
 
 For the sake of reference, the relative variations of Trigonomet- 
 ric lines, as above established, are exhibited in the following table. 
 
 Hi 
 
, (69i, and (70). 
 
 .iinati'd in the 
 e fCconJ I'roni 
 )i'e remain pos- 
 
 of Ti'i^ononu't- 
 bllowing tiiblo. 
 
 —2!)— 
 
 S,4 
 
 of 55, 
 
 Its (55i, (56) 
 
 the same point 
 are relating to; 
 Qua referred to 
 lity of the arc, 
 3d towards the 
 opposite diroc- 
 kdraut: 
 
 •J"Ar>I.E II. 
 
 Relative VAniATio:,"t! of AXCJL.vn Fum:tio;.'3 
 
 1 
 
 1. Qiiiul. 
 
 •-'. (^i....l. 
 
 3. (yMi 
 
 i. Qiiiid. 
 
 
 i Sin 
 
 + 
 
 + 
 
 
 
 
 Cos 
 
 -i- 
 
 — 
 
 
 + 
 
 
 Tun 
 
 + 
 
 -_ 
 
 + 
 
 
 
 Cot 
 
 -1- 
 
 
 
 + 
 
 
 
 Sec 
 
 H- 
 
 — 
 
 + ' 
 
 1 
 
 ^Cosec 
 
 + 
 
 -f 
 
 
 
 
 
 A\'rs 
 
 -1- 
 
 -j- 
 
 + 
 
 H- 
 
 
 Covers 
 
 + 
 
 + 
 
 4- 
 
 + 
 
 
 « 11 
 
 1 
 
 y 
 
-30— 
 
 Nnwcrical Variatiuns (*f Annnhtr FtinctldHn. 
 
 ]X'> 
 
 11; 
 
 
 Tlin variable angle, or the anglo of (liH'erent iimgnitudcs, which 
 angular functions aro icferrtMl to, may V)e concoivi-d ax generated 
 hy the levolutioD of a movable radiuH 6'/* I 
 round 6', from the fixed radius A<', in a 
 direction to the loft. It is manifest that as | 
 the angle decreaseM, the sine, also, decreas- 
 es, because Z*,^, is less than P^M, 
 
 When tlie rcivolving laclius coincides with | 
 CA, or the angle ))eflomes ft, PM - 0. 
 Therefore we have 
 
 sin 0' = |71)' 
 
 When the revolving r;idiu8 coincides with CJi, or the nngle be- 
 comes ylC^, a right angle, Pj»/ also coincides witli 67^, and is 
 equal to the radius = 1. Therefore, we have 
 
 sin 90° ■--- 1 |72| 
 Again, when the revolving radiiis proceeds in the second quad- 
 rant, the sine diminiMhos as the angle increases, becAuso P^Af^ia great- 
 er than P^M^ When the revolving radius coincides with CA^, 
 
 or the angle is equal to two right angles, PM = O.Theiefore we have 
 
 sin 180' = (73) 
 
 Reasoning in the same manner for the third and fourth quad' 
 rants, we shall find 
 
 sinQIO' = — 1 (74) 
 
 sin 360" = sin 0° = (75) 
 
 If now we observe the variations of cosine, we see evidently 
 
 that as the angle increases, the cosine decreases in the first quad. 
 
 rant, increases in the second, decreases in the third, and incraesea in 
 
 the fourth, and we have 
 
 cos 0° = C4 = 1 
 eo8 90° = 
 cos 180° = — 1 
 
 cos 270° = (79) 
 
 cos 360° = cos = 1 (80) 
 
 Thenegatire sign is employe* for sin 270° and cos 180°, be- 
 
 (76) 
 (77) 
 (78) 
 
?,]■ 
 
 lie nnRle he- 
 
 Ciiuao C'/?, is liulow .l.l., aud ('At is lo tlio loll yf /Hi . 
 ilunou it is hcoii, first, that sine 
 
 increases with + from to 1, iu tlio Ist ((iiadraul/ 
 <locroas''s witli -\- fi'oiu 1 iu in tho '2iid. " 
 increases with — from ti 1, iu tlie Std. " 
 decreases with — from 1 to 0, in the 4tli. " 
 Aud ;hat convoraely Cosine 
 
 docreas is witli + i^om 1 to 0, iu the 1st. 4U:idrant; 
 increases witlx — from to 1, in tlie 2ud. 
 decreases with — from 1 to 0, iu tlie 3rd. •' 
 increases with + from to 1, in tlio 4th. " 
 It is seen, secondly, that the limits of the variations being tho same 
 aud Ij for both Sine and Cosine in each of the four quadiaiits, it 
 follow^ that the values of iSine and Cosine of all arcs extending ; o 
 any quadrant are ini^luded between aud 1, therefore, tho numori- 
 calvalue of Sine and Cosine of any angle gioitur than 90 % is redu- 
 cible to either Sine or Cosine of any auglu not exceeding 90°. 
 
 This mo.'Jt important conclusion isillustii'ed iu the following 
 analysis: 
 
 Because Sine increases from to 1 between 0° and 90', in the same 
 jiroportion as Cosine decruasi.'s from 1 to in the same quadrant, it 
 follows that: 
 
 sin 1° = cos 89' 
 
 .sin 2° = cos 88° 
 
 and generally, sin a = cos (90° — <i) (81) 
 
 For the same reason, we have 
 
 cos a = sin (90° — a) (82) 
 
 that is; ike sine of an aiiijU is equal to the cosine of its comjjtt- 
 mimt; and the cosine of an awjle, is equal to the sine of its com- 
 jilement, 
 
 Because both Sine and Cosine decrease in the same proportion 
 from 1 to 0, the former between 90° and 180°, the latter between 
 (1° aud it 90°, it follows that 
 
 sin (90°) = cos 0° 
 
 i 
 
 4 
 
 M 
 
-92- 
 
 ■1 
 
 :t 
 
 
 siu (90' + l) = cosl° 
 siu(90°-|-2) = co8 2° ... 
 and generally, sin (90" -f a) = coh a 
 In the same way wii«obtain 
 
 COM (90° -\-a) = — Hin a 
 
 Because cos a = sin (90° + a), hj (88) 
 
 and cos a = sin (90° — a), by (82) 
 
 then sin (90° + a) = sin (90° — a) 
 
 but sin (90° -j- a) =8in [180°— (90° —a)], which is 
 
 evident, 
 
 therefore, sin [180° — (90° — a)] = sin (90° — a) 
 
 Substituting in these two supplementary angles a for (90* — a) 
 the supplemental relation remain th e same, and we have 
 
 
 or 
 
 
 Because 
 
 (83) 
 
 and 
 
 
 we have 
 
 (84) 
 
 Hence 
 
 
 or 
 
 
 but 
 
 
 Therefon 
 
 h is 
 
 
 sin (180° — o) = sino 
 
 (8fi) 
 
 that is. the sine of an angle ia equal to the sine of its supplement. 
 
 Because 
 
 and 
 
 we have 
 
 but 
 
 Therefore 
 
 Hence 
 
 Or 
 
 sin a = cos (90° + a), by (84) 
 sin a — cos (90°— a), by (81) 
 
 ■ cos (90° + a) = cos (90° 
 
 ■ a) 
 
 — cos (90° -I- a) = — cos [180° — (90 — a)] 
 
 — cos [180° — (90° ~a)]=C08( '^° — a) 
 
 — cos (180° — a) = cos a 
 cos (180° — a) =— cos a 
 
 (86) 
 
 That is, the cosine of an angle and the cosine of its supplement 
 are numerically equal, hut have unlike sign. 
 
 Since sine increases between 180° and 270° in the same propor- 
 tion as between 0° and 90°, it follows that 
 
 — sinl80° = 8iu0° 
 
 — sin 181° = sin 1° 
 
 — sin 182° = sin 2° 
 
 — sin (180° + a) = sin « 
 
 (87) 
 
 and generally, 
 or 
 
 sin (180° + o) = — sin a 
 And for the same reason, we have 
 
-33- 
 
 or 
 
 UocauHP 
 
 and 
 
 we have 
 
 Hence 
 
 or 
 
 but 
 
 Therefore 
 
 (88 1 
 
 ~ cos ilfO"-!""' — cof* 'I 
 
 COM llMO'-l-rO — — cos (I 
 
 Hinrt = — ftin (180° +a), by |87) 
 8in« = Hiu (180° — a), by <8B| 
 
 Hin (180° — r(i = — win |180° + «| 
 
 win (180°-f90° — rtl = — sin (180° — 90" + «i * 
 
 bin (270° — «) = — (in(9(<°-f a) 
 
 Hin (9;)' + '' =°^< ''. ''y (8Ji 
 sin (270°— rtl = — coHrt (89) 
 
 the same propor- 
 
 * Thli inftrcnoo rcstn on tlio princlpl(> Hint when the valiio nt one <|iuulruut 
 Ix added or Hiibtractcd from twoniiglCR, the relation of equality whlrh waHfotiiid 
 t'xlHtlng between the nrmerlcnl valucKofsome fUnrtlonsof the primitive an. 
 Klea flubslats between the rcmalnderR. For uHHiimo the urog Al*i, JJPv, Jiiv, Ail'4 
 All's Bil% Bil'i, AI'n t4> be all equal to one another. Ihoii IttveaNy to 
 prove geometrically, by the equality of trianglei, that 
 
 PiMi = P4M4 - I'm* = PiiMi 
 and l'»M' -■ PaMi = - PivMi -= PiMj 
 
 But P«M n Is the Kine of the arc A Ai Pi; and l':i m , Is the dno of tlw ftfo APj; Umro- 
 fore We have sin AAipn ^ hIii AP.i 
 Hiibtracting fi<im AAilV Ihoiiimdrniil I'.l'.i Uiivob 
 the arc Al'J whooe Htne U I'* M4 
 Hubtractlng from APa the qumlnint P Pa loaveH the 
 arc APi whoKe Kino In PiMi. 
 
 And since lUM* ^- PiMi, we hnve 
 Kin AP« - sin APi 
 that Is, If sin AAiPe =- sin APu 
 we have sin (AAiPn - 90^ = sin (AP.i - «)') 
 
 This holds true, If, Instead of diminishing 
 AAiPs by the quadrant PjP', wc Incrontie It by iho 
 <inadrant PePt; for PsMi ^ P4M4 -= PiMi 
 
 Again since PiMj ^ I'eMr, we have 
 
 gin AA1P7 -Kin AAiP« 
 
 From AA,P,8ubtraH the quadrant P.P:; and f.om AA>P«subtruct the qnadrun. 
 
 P4P., there remains AA,P-' whose sine is P,M.i and AP4 whoso sine Is P4A ' 
 
 B"t I';M4-=P4«4;nenco '' 
 
 The same holds tnie, If, lnstia-1 of (Kmlnl^hlng AA, Pa by the qiiiulrnnt iVP 
 
 In ca"r " "' '"^''' '"■■ ''*'" - '''''* ^'''" ■*"" "- p»ie isge:!., f,.; 
 
 The same result Is obtained by adding or subtracUng twoqnfldrant* oranv ..th..r 
 multiple of full quadrants, us Is evident. '^v^inftaranw, orany other 
 
 It must bo borne m mind that the equality relates only to the numerical »„l 
 ue, an not to the olgebrolc signs, which change aocordinu to X" Iw m . 
 ^ulcs the relative vorlatlons, as before established. "'""'""^ '" '"" '"" """ 
 
 1 
 
Ill 
 
 —34— 
 
 ]5y lullowing ji «iiiiilar procoiw wt* obtain 
 — con (270" — lit = Hin a 
 or COM (270° — «i = — nil! u (fjO) 
 
 Sinco l)oth Sini! and CoHinn docroaMu from 1 to (t, in tho Hunio 
 jtropoitiou, till* t'orintT Ix'twccn 270" and :l(!()°, and the latliilM... 
 t,w™n 0'^ and 9(»°, it I'oIIowh, olwervinKtlit' ride Ibr tlu! Si;,'ns tiiat 
 
 — Hin 270° = COM 0° 
 
 — «in 27r = 008 1° 
 
 — Hiu 272° =coH 2° 
 
 and generally — sin (270° +'«! = com u 
 
 Hin(270° 4-M = — coH(( 
 And for the same reason we have 
 
 coti (270° -f'«' = win <i 
 
 n> 
 
 (89) 
 
 Because cos *( — — sin (270° -f u , ]>y i!)l i 
 
 ""'l cos(i = — sin (270° — ,/i, l,y iKiJi 
 
 we have — >io (270° — «i = —sin (270'-|-rti 
 
 lionce — sin 1360° — «| = sin (180' -f ai, xec />fi ,,■ 33 
 
 but — sill (180°-f-a) = Hiu «, by H7) 
 
 thert'font —sin (3G0° — fl) = sin ^< 
 or 8in(360°— a) = — sin '< 
 
 By following a similar |irocess, we obi lin, obsorvin-,' tho conven- 
 tion for the signs, 
 
 coa(360'' — al. ..08<( ^l^^ 
 
 Sinco the movable radiu.i, after running through 360°, recom- 
 mences tho same revolution as before, we have, 
 
 sin (360°-f-al = sin « ,95, 
 
 (J)3) 
 
 cos (3G0° + «|= cos a 
 
 (96) 
 
 that is (he Sine and Cosine of an awjie vhirf, exceeds 360° are the 
 same as those of the excess above 300°. 
 
 The HincM bi-lng oquul.Ko will be tho i'(,8li)p, since 
 
 cos , 1 ;;i77: 
 
 Tho sines urnj cosines boing equal, so will l^ all the trlgononiolrln llnog. since 
 they are cxpresNoU In tfrnriH of sine mid cosine. 
 
 It Is evident that wlint Is said here of the nros Is nppllcnblo to the nnirle 
 which the arcs mcadiir*. ^ 
 
 V 
 
iuj,' the conveii- 
 
 h 360°, ivconi- 
 
 or 
 
 but 
 
 therefore 
 
 i9G| 
 (Is 3G0° aid the 
 
 Also 
 or 
 but 
 therefore 
 
 
 mctrlr lliiHH, Kincc 
 
 Also 
 
 iblo Ui tho angle 
 
 or 
 
 .1 p „ 
 
 Tho iiii|ioitaiit .Ic.liictioin wlii<'h |iiv.',',..I n.av I,c \n,H.Ml ^'.-ouio- 
 liimlly, Cor the satisluctioii ot the •'tu.lriii, l,_j t]„. uuuex..! ligiu.) 
 Hhui) llii' lire .1/', ii ~~ 
 iniiile ei|niil to ii; n\.A | 
 ih(^ arcs /?/». ///', 
 ,1, P„ .1,/'. II I', 
 B,P., AP^MvA\s\\Y- 
 
 |lUS('ll<M|Ulli \0 .' I' = It, 
 
 Thcrefoie, liy iviisti i;i'- 
 tioii, thetriiin;,'los.If 7', 
 
 /icp, nop, A,cj\ 
 
 A,< r, li^t'.^H L'P:^ 
 ACP, lire all equal. 
 Jlence, obscrviny tho 
 convention for tho iil- 
 jj'ebric si|,'ns,\ve Iiave 
 
 ~ P.M, =r'.i/, 
 or sin A P., = cos A P, 
 
 but yiP, = /iiy- 7?/\ = nn' — ffl 
 
 therefore sin DO" — wi = cos »< 82l 
 
 Also P,X>„ or its iMiual CM/, = P,.I/, 
 cos AP, = sin A Pi 
 
 AP, = An—flPr-W- a 
 cos (90° — ai = Pin a (81) 
 
 PM,= CM, 
 
 sin. I/", = cos ilP, 
 
 sin (90° -f*/ 1 = cos a (83| 
 
 - PjD, or its equal 03f, = P,M, 
 
 -cos AP,, = sinvlP, 
 therefo e — cos (90° -f </) = ein a 
 or cos 1 90° -f rt) = — gin a (S 1 ; 
 
 Also P,iV. = P, J/, 
 
 f* 
 
 II 
 
 i 
 
 
m 
 
 
 —30-- 
 
 \\t 
 
 t)'\ ' 
 
 1." 
 
 tf'i 
 
 it' 
 
 or sin AP^ = sin AI\ 
 
 l)iit Al\ = ABA, — A,l\ r= 180° _ a 
 
 therefore sin (180° — a) = sin a (85i 
 
 t 
 
 Also CM, = CM, 
 
 or . — cos APt = cos AP, 
 
 therefore — cos ( 1 80° — r? ) = cos a 
 
 or cos (180° — a) = — cos a |86| 
 
 -Also P,M, = P,M, 
 
 or — sin/lP, = sin 4P, 
 
 tut AP, = ABA, 4- A,P, = 180° + n 
 
 therefore — sin (180° -f a) = sin a 
 
 or 8in(180°4-a) = — sin4i87) 
 
 Also 
 
 CM, = CM, 
 
 that is . 
 
 -cos (180° -fa) = cos a 
 
 or 
 
 co^(180° 4- a) = — cos a (881 
 
 Also 
 
 PM = CM, 
 
 or 
 
 — sin AP^ = cos AP, 
 
 but 
 
 sin AP, = AA.By — B,P, = 270° — a 
 
 therefore' 
 
 — sin (270° — a) = cos a 
 
 or 
 
 sin(270° — <(l = — cosa (89) 
 
 Also 
 
 PeB,, or its equal CM, = P,J|, 
 
 or 
 
 — cos^P, = sin^P, 
 
 that is — 
 
 cos (270° — «) = sin a 
 
 or 
 
 cos (270° - al = — sin a (90) 
 
 Also 
 
 P,M,^ ^M, ' 
 
 or 
 
 — sin AP - cos^P, 
 
 kit 
 
 AP. = A A.n .A- n T> — OTAO 1 
 
 therefore — sin (270° + a) = cos a 
 or sin(270° + a) = — cosaOl) 
 
 Also P.D,, or its equal CM^ = p,M, 
 that is cos(270° + o) = sina 
 
—37- 
 
 Aiso p,M, = p,m; 
 
 or — smAA^Pi = sin APi 
 
 but AA,P, = AA,BA, — AP, = S(iO° — , 
 
 therefore — sin (360° — n) = sin a (94) 
 
 Also CM^ is the cosine of both Avl,P, and AP,; 
 
 therefore cos (360° — a) = cos a (94) 
 
 Substituting the above values of Sine and Cosine in 
 (20) (23) gives 
 
 ^ ^- sin 0° 
 
 tan 0° = = — =0 
 
 cos 0° 1 
 
 „^„ sin 90° 1 
 
 tan 90° = = — = 00 ■ 
 
 cos 90° 
 
 (11) (15) 
 
 1971 
 (98) 
 
 % 
 
 .1 
 
 
 , ,„„_ sm 180° 
 
 tan 180° „ = =0 
 
 cos 180° 1 
 
 (99) 
 
 ^^.^^._sin270°_l 
 cos 270° 
 
 (100) 
 
 tan 360° - ^ '''' - ^ = o 
 cos 360^ 1 
 
 (101) 
 
 cos O"" 1 
 
 cot 0° — . ^ , = =00 
 
 sin 0' 
 
 (102) 
 
 , - ._ cos 90° 
 cot 90° — -^_- = =0 
 sin 90° 1 
 
 (103) 
 
 ^,_-o cos 180° 1 
 ''' ''' - sin 180° = = " . 
 
 (104) 
 
 cot 270° - -^ ^,^^«: = ^ _0 
 sin v'70° 1 
 
 ■ (105) ' 
 
 C0t360°=''°«^«f:= ' =00 
 sin 360" 
 
 (106) 
 
 '■ * 
 
 
 • 
 
 
 I 
 
 ll 
 
 "Hi 
 
 i 
 

 '.*- 
 
 I ; 
 
 —38— 
 
 fir 
 
 III 
 
 1 1 
 
 cos 0^ 1 
 
 sec 90= 
 
 1 
 
 cos 90'' 
 
 8f 180° =-- 
 
 1 1 
 
 sec 270° = 
 
 cos 180° 
 1 
 
 Oc 
 
 
 COS 270" 
 
 _ }__ 
 
 sec 360° = 
 
 1 
 
 cos 360° 1 
 
 
 ilOTi 
 (108) 
 1 1091 
 (110) 
 
 dill 
 
 cosec 0° 
 
 sin 
 
 (112) 
 
 cosec 90° = -T 
 
 1 
 
 cosec 180° 
 
 sin 90-' 
 1 
 
 = 1 
 
 1 
 
 sin ISO-' 
 
 cosec 270° = - 
 
 
 sin 270- — 1 
 
 qoBec 360° = ■ .,-,.o = — - 3C 
 sin 360 
 
 ai3i 
 
 lll4i 
 
 (115) 
 (116i 
 
 I ! 
 
 sin(90°— A) cos a ^ ,,_^ 
 
 tan (90° — a\= —^—5 = —. = fot n (117) 
 
 cos ^90° — a) sin a 
 
 „ . sin (90° 4- a) cos a . ^ n a\ 
 
 tan (90° + a = ^ — ~ - = -. — = —cot a 1II8 
 
 wn(,»u -r co8(90° + rt) —sin a 
 
 .:*; 
 
cot a ill7) 
 
 = — cotrt 1.118) 
 
 tan 1 180°— a) = 
 
 tarn 180°+ a]= 
 
 9in(180°— «) __ siu a 
 cos (180°— a) ~ ^^^^cos « 
 
 sin 1180° + «) — sinrt 
 
 = — tana |11()| 
 
 = tan a 
 
 (120) 
 
 tani270^_«i 
 tan (270° + ai 
 tan (360° — r/ 1 
 tan f360° + a) 
 cot (90° — a] z 
 
 cot (90° 4-rt)= 
 cot(180° — a|=: 
 oot (180°-f-rt) = 
 
 coH(180°+o) — cosrt 
 
 ain (270° — g) — cosrt 
 
 n^o lOTfio T ~ : — = cot a (121\ 
 
 cos(i70°— a — amo "•'^' 
 
 COS a 
 
 __ sin (270° -}-rt| _ 
 cos(270°~+^~ sh:;, 
 
 ^ sin (360° — 01 _ _sin a 
 cos (360°— a I c^^ 
 
 ^ ^mj360°j4-«) _ sin a __ 
 
 cos (360° + «r "^a - *'''" « 
 
 cos 1 90° — a) sinrt 
 
 = — cot a (122l 
 
 = —tan a (123l 
 
 CO." (7 
 
 sin i90°_,( 
 
 c os (90° -4 - g; —sinrt 
 sini90°-f:rt) ~^a 
 
 ~ fan <i 
 
 = — tana 
 
 cos a 
 
 C 08(180° — ,?) _ 
 
 sin (180° -"Tii = "di^ = - cot rt 
 
 co8(18 0° + rt ) _ — cosa 
 sin (180° -\- n] ~ —ain a 
 
 = oot a 
 
 cot,270°-ai=-^°i(^«:=I^ = 
 sin (270° — a) 
 
 — sin a 
 
 — COS a 
 
 = tnn n 
 
 cot(270° +.a) = ^«(!IO;±^t ^ Ji^_ _ ^^ ^ 
 
 ^ 8iD(270° + rt) -C08«~ ^°" 
 
 (124) 
 (12r)) 
 
 (12(J) 
 (1271 
 
 (1281 
 
 (129) 
 (130) 
 
 
 1 
 
 l!^::l 
 
i 
 
 -40— 
 
 :i\ 
 
 I:v,i, 
 
 
 cot (360° — a) 
 cot (360° + a) 
 
 068(360° — a) COB a 
 
 sin (360 °_ a) —sina 
 
 _ cos (360° -\-n) _ cos a 
 ~~ sin (360° -\-a\'~ sin a 
 
 = —cot a (131) 
 
 cot a (1321 
 
 sec (90° — rt) = 
 
 1 
 
 1 
 
 cos (90° — a] sin a 
 
 = cosec n (133) 
 
 sec (90° + rti = rr-;^o-r— : = — -- _ = — cosoic u (134 1 
 
 co8(90°-fal — sin a 
 
 sec (180° — a) = 
 sec (180° + ffli = 
 sec (270°— rt) = 
 
 008(180" — a) — C08c( 
 
 - = — spc « (135) 
 
 1 
 
 008 (180'^ -f- a) — costj 
 
 l^ 1 
 
 COS (270° — o) ~ — sin rt 
 
 = — seca (136) 
 
 • cosec rt (137' 
 
 1 
 
 sec(270°-f ffl = __^,^^^ ^ 
 sec (360° —a) = 
 
 c^s |270°-|-a) sin a 
 1 1 
 
 1 » 1 
 
 cosoc a (138) 
 
 cos (360° — a) cos a 
 
 = sec a (139) 
 
 cosec (18( 
 
 cosec (18( 
 
 co8ec(270 
 
 cosec (27(1 
 
 oosec (36(1 
 
 cosec (36( 
 
 
 
 (140) 
 
 i' 
 
 _ , C09(360*^ + rj) cos a 
 
 , 
 
 
 (141) 
 
 
 sin (90° — a) cos a 
 
 ; . ;r 
 
 
 (142) 
 
 
 sin (90° -4- a) cos a 
 
 • 
 
 ■ ■-:(' 
 
 • 
 
 
 
 
 
 ( 
 
 * ' ...-". 
 
 
 In certa 
 to the angi 
 angle to be 
 fixed point 
 and the an; 
 positive, or 
 we suppose 
 revolution 
 the oppo'it 
 jj considerc 
 
 It is ma 
 ative have 
 coiresponf 
 
—41- 
 
 l'i» 
 
 cot a (131) 
 
 !otn (132) 
 
 sec a (133) 
 
 -coscfe a (1341 
 
 — src rt (1351 
 
 — sec a (136) 
 : — oosecrt 1 137' 
 
 cO8ec(l80° — a)= - 
 
 1 
 
 1 
 
 sin (180°— .a) sin o 
 
 cosec a (143) 
 
 co8ec(ia0° + al = 
 
 co8ec(270°— a) = 
 
 1 
 
 1 
 
 ain(180°+rtl —sin it 
 
 — cosec a (144) 
 
 1 
 
 8in(270° — «) —cos a 
 
 cosec (270° + a) = 
 
 cosec (360"" — a) = 
 
 1 
 
 1 
 
 8m(270"-|-rti — cosrt 
 
 = — sec n (145) 
 
 — — aec a il46| 
 
 sin (360'— ai 
 
 mxx a 
 
 = — cosec « (147) 
 
 cosec (360°-f-rt) = 
 
 1 
 
 1 
 
 ain(360'4-<" s''^ '* 
 
 = cosec rt (148) 
 
 ^^■\ 
 
 „ '^n:^ 
 
 V '" 
 
 
 fvV 
 
 i 
 
 ospc a (138 1 
 
 lec a (139) 
 
 sec a (140) 
 
 !C a \U\) 
 
 Bco (142) 
 
 Flnctions op Negative Anglbs. 
 
 In certain investigations it is convenient to give different signs 
 to the angl»8 themselves. "We have hitherto supposed the variable 
 angle to be generated by the revolution of a straight line round a 
 fixed point, in a direction from right to left; p 
 
 and the angles thus formed are considered as ^ -— ''^ 
 
 positive, or have the sign -|-, as ^4 CP. If now ,^''' 
 
 we suppose an angle to be generated by the ^" ^;^~ 
 revolution of the movable line from CA in -- 
 
 tlie oppo-'ite direction, the angle thus formed ^"^/" 
 
 i J considered as negative, or has the sign — , as ACP'. 
 
 It is manifest that the functions of an angle considered as neg- 
 ative have the same absolute magnitude as they would have for the 
 corresponding poiilive angle, or the angle formed by revolving 
 
 
\\W 
 
 -42— 
 
 i 
 
 tlic nioviililo Hue in tlu- positive ilirection until it readies the po- 
 tiition wliich it has in ihn fi^'urc. Tlieru may he found a diflt-'rcnct! 
 only in tlie algi'l)i'aic signs. Thus, suppose tlie negative angle ACP' 
 to he denoted hy a', and the corresponding positive niiglijhy O' we 
 have evidently, the angle heing in the fourth ijuadiMnt, the sine 
 negative and the cosine positive; that is 
 
 siin ( — a'l = sin (J lUOi 
 
 cw i — (('i = eos« iir>()i 
 
 wlience by ( 1 1 1 and (15i 
 
 , tin I — «/') — sin a 
 
 tan ( — </ I = = - " = — tan a ilAli 
 
 COR I 
 
 ■ (/ I 
 
 cos a 
 
 cot I — (() = 
 
 COS ( — a'\ 
 
 COS (/ 
 
 sin I 
 
 ■ sin ti 
 
 see , — «'l := 
 
 cosec I — a'l = 
 
 1 
 
 COS I — ri'l 
 
 1 
 
 sin ( — o'l 
 
 1 
 
 COS ,1 
 
 1 _ 
 
 — sin n 
 
 = — cot a il.')2i 
 ■lec « |]53i 
 
 = — cosec n tl54i 
 
 Hence, tfic sine, tangenf, aitaiUjinl, ami euserant, of a negatine 
 anijle, have the sign o])posiieto fhntofthe. sinr, tangent, rotanffe7it, 
 and cosfcant, cj the torret,]otilitn; imstitive angle: and the cosine 
 and secant hare the srnie sign in both the negative and the corres- 
 ponding positive. 
 
 For the sake of reference, we collect the preceding results in the 
 following tiible. • 
 
 ■f- :i 
 
—43- 
 
 achea tlie pu- 
 1(1 11 difli'vencti 
 vo an','lo ACP' 
 ui,c,'l(jl)y fl; wo 
 .'.int, the Hint' 
 
 .1491 
 
 anoi 
 
 -tan II 1 151 1 
 
 - cot « (1521 
 
 il53i 
 
 -cosec n (154i 
 
 t, of a negative 
 lent, cotangent, 
 and the cosine 
 and Ihecorres- 
 
 jj results in the 
 
 TABLE 111. 
 Numerical VARiATtoNs of Angulau Functions. 
 1, — Value of Functiom of Qwtdronfal anf/les. 
 
 •■fti 
 
 sin 0° = 
 
 
 COS 
 
 (p - 1 
 
 8in 90' = 1 
 
 
 COS 
 
 90° = 
 
 .sin 180° = 
 
 • 
 
 COS 
 
 180° = — 1 
 
 .sin 270-^ = — 
 
 1 
 
 COS 
 
 270° = 
 
 8in 360° -- 
 
 
 COS 
 
 360° = 1 
 
 tan 0° = 
 
 
 cot 
 
 0°= * 
 
 tan 00" = 00 
 
 
 cot 
 
 90" = 
 
 tan 180° = 
 
 
 oot 
 
 180° -— X 
 
 tan 270° = — 
 
 Vj 
 
 cot 
 
 270° = 
 
 tan 360° = 
 
 
 cot 
 
 360° = * 
 
 sec 0° = 1 
 sec 90° = 00 
 sec 180° = — 1 
 sec 270° = ao 
 sec 360° = 1 
 
 cosec 0° = Qo 
 cosec 90° = 1 
 cosec 180° = Gc 
 cosec 270° = — 1 
 cosec 360° = rr 
 
 2. — Reduction of the Functions of all Angles to correspond iii'j 
 Functions of an Angle less than 90°. 
 
 sin (90°- «i = 
 
 co< 
 
 a 
 
 cos (90' — a) = 
 
 sin 
 
 a 
 
 .sin (90° + ai = 
 
 cos 
 
 a 
 
 cos (90'-frt) = 
 
 — sin 
 
 a 
 
 »in (180° — ai = 
 
 ^ sin 
 
 a 
 
 COS (180° — a) = 
 
 — cos 
 
 a 
 
 sin 1180° -fa) = 
 
 — sin 
 
 a 
 
 co.s il80°-f-al = 
 
 — co^ 
 
 a 
 
 sin (270° — al = 
 
 — cos 
 
 a 
 
 COS (£70° — a) r= 
 
 — sin 
 
 (1 
 
 .sin (270° -fa) = 
 
 — cos 
 
 a 
 
 cos (270°+ a) = 
 
 sin 
 
 1). 
 
 sin (360° — rt) = 
 
 — sin 
 
 '1 
 
 COS (360" — «i = 
 
 cos 
 
 a 
 
 sin (360°-f-«i = 
 
 sin 
 
 a 
 
 cos (360'-f rti = 
 
 cos 
 
 a 
 
 tan (90° — ai = 
 
 cot 
 
 a 
 
 cot lOO' — ai = 
 
 tan 
 
 a 
 
 tan (90° -f n) = 
 
 — cot 
 
 a 
 
 cot i90''-f,/i — 
 
 — tail 
 
 a 
 
 
 [ 
 
 m 
 
 I 
 
1 
 
 —14- 
 
 lau ,180'— (n = 
 
 — tall 
 
 a 
 
 cot (180° _(r- 
 
 — cot 
 
 (( 
 
 tun (180°4-<f) = 
 
 tan 
 
 (t 
 
 cot (18j° + ^' = 
 
 cot 
 
 « 
 
 tan (270° — rt) = 
 
 cot 
 
 it 
 
 cot (270° — rtl = 
 
 tan 
 
 a 
 
 tan(270° + rt* = 
 
 — cot 
 
 a 
 
 cot (270° + «) = 
 
 — tan 
 
 (I 
 
 tan 1 360° — .71 = 
 
 — tan 
 
 <i 
 
 cot (360'' — rtl = 
 
 — cot 
 
 It 
 
 tan (360°-|-«| = 
 
 tan 
 
 ft 
 
 cot (360^ + rt) = 
 
 cot 
 
 II 
 
 ace (90°— a) = 
 
 cosec a 
 
 otsec (90° — a) = 
 
 S(,'C 
 
 a 
 
 sec ,90° + (f| = 
 
 — cosec a 
 
 cosec (90° + a) = 
 
 sec 
 
 a 
 
 sec (180°— a) = 
 
 — sec 
 
 a 
 
 cosec (180°— a) = 
 
 cosec 
 
 a 
 
 sec (180° + a) = 
 
 — sec 
 
 a 
 
 cosec !l80°-|-rt) = 
 
 — cosec 
 
 (1 
 
 dec (270°— a| = 
 
 — cosec a 
 
 cosec |270° — ni = 
 
 — sec 
 
 a 
 
 soc (270° + <-j) = 
 
 cosec 
 
 a 
 
 co-.oc (270° + «i = 
 
 — sec 
 
 (1 
 
 sec (360° — rt) = 
 
 soc 
 
 a 
 
 cosec 1360'— (/) = 
 
 — cosec 
 
 a 
 
 spc i360°+^f> = 
 
 soc 
 
 ,t 
 
 cosec (300° + «i = 
 
 cosec 
 
 a 
 
 Thus we see that Sine and Cosine iniy liavc any value between 
 — 1 and +1; Tangent and Cotangent may have any value be- 
 tween and oc; Secant nnd Cosecint may have any value V^tween 
 1 and oc. 
 
 Exercises. 
 
 1. To determine the value of the. angular functions of 45° 
 By the equation [a], paife 13, we have 
 sin^ -\- cos'- = 1 
 but sin 45° = cos 45°, by (82) 
 
 therefore sin' 45° = cos' 45° = ^ 
 
 sin 45' = -^ = 0.707 U)68. 
 
 hence 
 
 and 
 
 V2 
 cos 45° = -\ = 0.7071068. 
 
 whence 
 
 •„ sin 45° , 
 
 tan 45° = -^ = 1 
 
 cos 45 
 
*< 
 
 cot 
 
 ii 
 
 cot 
 
 a 
 
 tiin 
 
 a 
 
 tan 
 
 ii 
 
 ■cot 
 
 It 
 
 cot 
 
 II 
 
 si'C 
 
 a 
 
 sec 
 
 a 
 
 C08CC 
 
 a 
 
 ■ coaec 
 
 <i 
 
 -sec 
 
 n 
 
 -soc 
 
 It 
 
 - C08CC 
 
 a 
 
 cosec 
 
 a 
 
 lie l)et\voen 
 
 V value be- 
 
 due V^tween 
 
 4r)°. 
 
 _-45_ 
 
 cot 45° = T— .77= 1 
 Hin 45° 
 
 1 
 
 sec 46 = 
 
 00800 45 = -: 
 
 cos 45° 
 1 
 
 Hill 45° 
 
 = ^2= 1.4142136 
 
 V2= 1.4142130, 
 
 vers 45' = 1 —cos 45' = 1 = 0.2928932.... 
 
 covers 45° = 1 — sin 45' = 1 
 
 V: 
 
 0.2928932.... 
 
 2. To determine the value of Angular Functions of 30°. 
 
 Since the sine of an arc is perpendicular to the diameter pass- 
 in" through one extremity of the arc, it follows evidently that the 
 sine of an arc is always equal to half the chord of double the arc, as 
 seen in the figure, page 35, where P, M., the sine of the arc AP^ 
 is the half of P^P.,, the chord of tlio arc PiAP^, double of the arc 
 i4P,. But it is proved in geometry that the chord of one sixth of 
 the circumference of a circle is equal to the radius of tluit circle. 
 Therefore, the radius being assumed, as unit.v, we liavc 
 sin 30° = J, whence 
 
 cos 
 
 30° = V[l — sin' 30°]= V[l — i] = '^| = 0.86C0254.... 
 
 I 
 
 M 
 
 U^i 
 
 , ' ii 
 
 J 
 
 ^''"^^ = co^3iF-*--2- 
 
 ./3 
 
 = 0.5773502..., 
 
 cot 30° = 
 
 _ cos 30° _ ^/3 _^ , _ g_ 
 
 sin 30° 
 
 i.J= ^3 = 1.7320508, 
 

 —46- 
 
 ■1 
 
 It":?' 
 
 H(.f 30° -- 
 
 coM'c 30'' 
 
 1 _ -2 
 
 COS 30° ~ y^3 
 
 -J- =2 
 8in 30° 
 
 = I .151701 
 
 V3 
 
 3. 
 
 vors 30° = 1 —cos 30° = 1 - - "~ = 0.1 330740 
 
 covers 30° = 1 — sin 30^ = h 
 To ilet ermine the value of AiKjular Finirtiohs of G0°. 
 
 Since 00° is the complement of 30°, we have 
 sin 60° = cos 30° = .80o0254.... 
 cos 00° = sin 30° = * 
 tan 00° = cot 30° = 1 7320508..,. 
 cot 00° = tan 30° = ,5773r)02.... 
 and so on. ^ 
 
 4. Ti> iktcrmiiiethv rahie of Aii(/ular Finirtiom of 110''. 
 
 Since 120° = 90° + 30° 
 then sin 120° = cos 30° = .8000254.... 
 cos 120° = — sin 30° =: — J 
 tan 120° = —cot 30° = — 1 .7320508.... 
 cot 120° = — Um 30° = — .5773502 
 and so on. 
 
 It i.s manifest tliat in the siinu! way, we may obtain, at once, the 
 angular functions of 135°, 150°, 210°, 225°, 240°, etc., which 
 are respectively equal to 90° + 45°, 90° + 00°, 180° + 30°, 180° 
 + 45M80°4-C0°.... 
 
 
CHAPTKR IV. 
 
 AxOlt/^ll F'oXCTIOXH ISVOIAIXG MoRK THAN ONE \'.rilAl)I,E. 
 
 FuuJaiiu'iUiil Pioiio-iition iinil Foniiulu. 
 
 The niiK' of till .^11 m cf (wo om/li-x or ,ircM, ix equal in the sum 
 "t llif /ridticf of til,' nine of (hfjir.st into thu cosine of the secoinl, 
 iind till' /,/v ihict of till- ro.-iiir ,,/ tli<-jir.it Into tint nine of the sect/nd. 
 
 Tlli^^ proposition u ciiilccl funiliiini'iitii! on account of its gnmt 
 iiniiortancc, as hcinj;, to^cthor witli tlic t'linilanicntai fornnihu f,'iv- 
 en in Chapter II. and rolatiuK to functions of a ain','l(; angle, tlm h\wi\ 
 of the. whole trigonoiiietncil scii'iicc 
 
 Thi.s proposition will ho proved f,'ooiiietrically; and wo will noxt 
 proceed n analytical invt.stif^atiou 
 
 Let the arc A P 
 = A ; and tht* arc 
 PT) = //; then 
 the arc AD = 
 AP-\-PD= A 
 -4- B. On CA let 
 fall th<' ])erp<'n- 
 iliculars / O, TL, 
 l>M. ])iaw tho 
 radius CP; ako 
 DT perpendicu- 
 lar on VP. and 
 
 ■{ 
 
\f 
 
 -48- 
 
 y.S' jK^i'iii'iKlicular on UM. Tlum wt- Imve, by dt liiiilioii !//«(/<• 8i: 
 
 PO = Dill A 
 
 (!0 = coH i4 
 
 1)T= Hill B 
 
 CT = COH Ji 
 
 DM = sin A1J=: mi \A + H^ 
 By conHtruction it iH Kvideut timt. tlie triaiit^lus ('FO, CTL, ami 
 TlJHtm- oiiiiiliir, tlieroJore; tultiiig tlin ladiiw us unity, w« hivu 
 
 or 
 
 AIho 
 
 or 
 
 from [A] w»! obtain 
 
 f'roui [/] wo obtain 
 
 Adding 
 
 but 
 
 UP : PO :: CT: TL 
 1 : «in /I .• ; C08 B : TL [/.J 
 aP : r'O : : /)'/ ; 7>.S 
 1 : cos i4 : : nin U : 1>S [/J 
 
 7*7/ = sin A com Z^ 
 DS = 008 i4 liin Ji 
 TL + />.S = Hin /I coH « 4- cos /I sin li 
 TL + DS = MS +Dii= DM = sin \A + H\ 
 
 Therefore sin iA -{- B, = sin A con W -f- ooh i4 sin /V 
 
 which in the funiltiniKntal formula involving two angli-s. 
 
 Angular Funt'tlom of [A -j- 7^) H'ltl \A 
 KeHuming the last formula: 
 sin (A-\- B) = sin A con B-\- cos A s'n B 
 
 B). 
 
 llfiS) 
 
 sin [A—B] = sin {\m° — [A — B)^, by (HSi 
 
 = «in [(180° — i^li + ^], which is evident. 
 
 = 8inU80° — .4) C08i3-fc08 1180" — iil sin B, by 
 
 (166) 
 = sin A coH B — los il tn h (166) 
 
 .by substituting for sin (180° — A) and cos il80^ — A\ their 
 values as foxind in (85) and (86). 
 
 cos (A-\-B\- sin [90° 4- (^ + ^']. ''Y t«3) 
 
 = sin [|90° + i4l + /?], whish is evident. 
 
 = 8in (9C°-t--«4) co8.B-f coB(90° + .4)sini?, (165) 
 
—49— 
 
 y 
 
 — COH A cos I}-- sin i4 sin fl, tty (83i and (84 1. (nSi 
 
 cos /I — /il = — coH [180' — (i4 — //i], hy i86i 
 
 = — COH [|1H0° — yll + H], which is eviilfnt. 
 
 =-- — [co8il80° — /tj COM n-Hin ilHO' — /li sin //], 
 
 by (1571 
 = -.cos(180° — i4i cos /y + «in ] HO" ~ A) n'm li 
 = coH A cox /^ -f- «'" A niii //, by 8Ci and 185 , i lfl8| 
 
 . . Hill {A -{• li) , ,, 
 taui^-f 7^1 = , ; ,,,hy ,11) 
 
 cox >A + Hi 
 
 riiii A uoH H -\-coi A hIji // 
 cox A coH H — Mill A m\ Ji' 
 
 by il.'55, and (16Ci 
 
 dividing niiniorator and dpnoniinilorby cox A cox li: 
 
 sin A COS li cox /I sin // 
 
 . cox A cos /i cos A cos // 
 
 cos i4 cox Ji sin /f sin .t 
 
 cos A cox /y cos ,1 cox y* 
 
 I 
 
 tan yl -f tan W 
 i — tan A tan 7? 
 
 (Ift9) 
 
 ^ „ Bin (.4 — Z?l , , , 
 
 tan (i4 — 7?) = rr, by dli 
 
 cos {A — Ji) •' 
 
 sin yl cos ^ — cos A sin li 
 cos A cos 7/ -|- fin ^t siu 77 
 
 Proceeding as bofore. 
 tan A — tan 5 
 
 1 -1- tan yl tan B 
 
 (160) 
 

 —50— 
 
 coa {A-\-n, 
 
 cot 1^ 4-7ii=: ----- - by (15) 
 
 Hin '^ -\- Jl< 
 
 cos ^ COS /y — sin ^ sin 7? 
 
 sin A cos yy -|- cos A sin 7; 
 
 dividing iiuiii. and den by sin A sin B, iindrednc- 
 ins; 
 
 cot A cot Ji — 1 
 
 cot Jfy -f cot A 
 
 ^ ^ ,, cos {A — yyi , 
 
 cot {A — JJ\ — , by ;15) 
 
 sin 1^ — Jj] 
 
 cos A cos Ji + sin A sin 7i 
 sin A cos Ji — cos A sin JJ 
 * 
 Proceeding us before: 
 
 cot ^ cot /;-f 1 
 
 cot 7y — cat A 
 
 (161) 
 
 ll62) 
 
 soc ^ -f 7ii = • •-•-, by (20) 
 
 cos A + /ii 
 
 cos A cos 7i — :;in A sin /y 
 
 •I. 
 
 Multiplying num. and den. by sec A sec JJ: 
 
 si'c A sec B 
 1 — tan i4 tan .fl 
 
 (163) 
 
 Multiplying num. and den. of ^l63) by cosec A co- 
 sec B, and observing that by (10) and (3^). tan 
 
 X cosec = sec: 
 
 11- *|!i 
 
n 
 
 —51- 
 
 
 sec A sec B coaec A coaec .B 
 cosec A cosecTB — seciisecS 
 
 (164) 
 
 *-r: ,| 
 
 sec {A — B) = 
 
 13, iind re<luc- 
 
 008 {A — B) 
 
 cos A cos J3+ sin /I sin fi 
 
 Proceeding as before: 
 sec A Bed B 
 
 l+tani4 \&uB 
 
 fiosec (i4 + 5) = 
 
 see A sec 5 cosec il cosec B 
 cosec ^ cosec B-\-Be(i A sec J? 
 
 1 
 
 Bin {A -I- 5) 
 
 5r» ^y (231 
 
 1 
 
 (165) 
 
 (166,) 
 
 sin A cos B -^ cos ^ sin ^ 
 
 Multiplying num. i.nd den. by cosec A 
 and reducing: 
 
 coaec A sec B 
 
 \ -\- eoi A i&n B 
 
 Multiplying num. and den. of (167) by sec 
 sec 5, and observing tlm.t, by (5) and (40), 
 sec = cosec: 
 
 sec A sec B cosec A cdsec B 
 ' sec A cosec B ^eecB cosec A 
 
 sec B, 
 
 (167) 
 
 A co- 
 cot X 
 
 (168) 
 
 I 
 
 » 
 

 :3 
 
 •li 
 
 ?-; 
 
 —62— 
 
 8in [A — H) 
 
 sin /I cos 5 — cos A sin J5 
 
 Proceeding as before: 
 
 cosec ^ sec i? 
 " 1 — cot X tan JJ 
 
 sec A sec B cosec ^4 cosec B 
 
 sec i4 cosec B — sec if cosec A 
 Parliculur Case of 46° 
 Let A = 45°; then wo obtain: 
 cos J3 + sin J? 
 
 sin (46° + J5) = 
 cos 145° + /?! = 
 tan i46°4-/?l = 
 
 
 Dec (45° ■ 
 
 
 cosec (46° - 
 
 (169) 
 
 
 
 Since (4{ 
 
 (170) 
 
 ein (45° - 
 
 cos (46° - 
 
 tan (45°- 
 
 And so 
 
 cot (45° + 5) = 
 
 V2 
 cos B — sin £ 
 
 cos B -f- sin B 
 cos B — sin ff 
 
 Dividing num. and den. by cos B or sin 
 
 l+Jtan_B 
 I — tan JB 
 
 cot.g4-l ! 
 
 cot 5 — 1 
 
 cos B — sin J5 
 
 (ITl) 
 
 (172) 
 
 073) 
 
 (174) 
 
 cos J5 + sin 5 
 
-63— 
 
 _ cot ii — 1 
 
 ~ cot £ 4- 1 
 
 «ec (45° + B) = .y^. ^ 
 
 cos J5 — sin if 
 
 V2 
 
 cosec (45° + B] = 
 
 cos 2? -|- sin ^ 
 
 (176) 
 
 (177) 
 (178) 
 
 Since (45° + -S) and (45° — B) are complementary, we have: 
 
 sin (45° — Bl = cos (45°-4- B\ = (172) 
 cos (45° — £) = 8in(46°-f -B) =(171) 
 tan (45° — B) = cot (45° + B) = (175) or (176) 
 And so on. 
 
 Angular Fwictions of 2 A 
 
 Making B — A \n the expressions for [A + .6), we obtain: 
 
 «in 2 A = sin (A -\- A) = «in A cos A + cos A sin A 
 
 = 2 sin A cos /I (179) 
 
 .- 2 sin ^ ^ [1 — sin* A^ (180) 
 
 = 2 cos J v' [1 — cos* A'\ (181) 
 
 cos 2 j4 = cos [A-\- A^ 
 
 = cos* ,4 — sin* A (182) 
 
 = 2 cos' .4 — 1 (183) 
 
 = 1 — 2 sin* A (184) 
 
 cos* A — sin^ A i. r -, 
 
 ~ C08* A 4- sin* A ' "''' "'?'"''*''" [«]'.P«^« 13.- 
 
 Dividing mim. and don. by cos* A: 
 
 
 ii'i 
 
 1 1 
 
 .^'■■-ri 
 
 , -:t 
 
 i 
 
 ;ivj, 
 
 ■if' 
 
*-84- 
 
 1— tan'yt 
 
 1 + tan* ^ 
 
 tan 2 A =im(A-\-A) = 
 
 2 tan^ 
 
 tan A -\- tan A 
 1 — tan 4 tan i4 
 
 1 — tanM 
 
 Dividing num. and den. by tan A: 
 2 
 
 cot A — Un A 
 
 cot 2A = CQt{A-\-A)- 
 
 cot« A — 1 
 
 cot A cot A — 1 
 cot A -^ cot A 
 
 2 coti4 
 Dividing nnm. and den. by cot A: 
 
 cot A — tan il 
 
 (185) 
 
 (186) 
 
 (187) 
 
 (188) 
 
 (189) 
 
 cosec 2 A = 
 
 „ . ^ , . sec A Bee A 
 
 sec 2 A = sec {A + A) = — —, by (163) 
 
 1 — tan i4 tan il ' ^ ' 
 
 sec* A 
 
 1— tan»i4 
 
 and by the equation [A], page 15 
 
 l + tan»il 
 1 — tan»i4 
 
 (190) 
 
 (191) 
 
—55— 
 
 
 163) 
 
 Also, by (164) 
 sec* A cosec* A 
 
 cosec' A — sec* A 
 
 cosec 2 A = lin 2 .1 = 
 
 1 
 
 1 
 
 2 sin A cos A 
 
 , which is evident . 
 
 (193> 
 
 2 sin* A. cm A 
 sin il 
 
 Dividing num. and den. by sin* A ;. 
 
 _ cost'c* A _\-{- cot* A 
 2 cot .4" 2cotil 
 
 Also, by (168): 
 
 sec* A cosec' A 
 2 sec A cosec ^ 
 
 Dividing num. and den. by sec A cosec A '■ 
 
 sec /4 cospc A 
 " 2 
 
 AiKjUlar Functions of ZA . 
 
 (193) 
 
 (194) 
 
 (195) 
 
 Sin 3 -4 = sin (2 i4 + ^ I = sin 2 .4 cos .(4 + cos 2 /I sin .4 
 = 2 sin ^ cos* .4 + (1 — 2 sin" .4) sin A 
 = 2 sin (1 — 8in*i4)4-(l — 2 sin" ^) sin i4 
 - 3 sin ^1 — 4 sin' 4 (196) 
 
 (Jos 3 i4 = cos (2 .4 -j- ^ ) = cos 2 i4 cos i4 — sin 2 il sin il 
 = (2 cos' A — 1) cos i4 — 2 cos .4 sin' A 
 = 12 cos'/l — 1) cos il — 2 cos^ (1 — cosM) 
 
 I 
 
m 
 
 —56- 
 
 3= 4 cos'' A — 3 cos i4 
 
 t&n 2 A -{■ tan A 
 
 |197. 
 
 tan 3 yl = t&n r2 A + A) = 
 
 1 — tun 2 A tan yl 
 
 2 tan yl , , . 
 
 — + tan A 
 
 1 — tan'yl ^ 
 
 1 
 
 2 tun A 
 
 , by il8Gi 
 
 1— tanM 
 
 tan A 
 
 Multiplying num. and lien, by 1 — tan" A, and re- 
 ducing : 
 
 _ 3 tan .4 — tan ' A 
 ~ 1—3 tan' A 
 
 Dividing num. and den by tan A : 
 
 3 — tan-il 
 
 il08) 
 
 cot il — 3 tan A 
 
 cot 3. 1 = cot(2A + A) = 
 
 <Hi9) 
 
 cot 2 X cot A- — 1 
 
 cot 2 A-\- cot A 
 
 cot' A — 1 
 2 cot A 
 
 .cot A — 1 
 
 = cotM — 1 
 
 , by (187, 
 
 -\- cot A 
 
 2 cot i4 
 Multiplying num. and den. by 2cotil, and reducing: 
 
 _C^tM--^COt^ ^^^^ 
 
 3 cot' -4 — 1 
 
 f;iD 4 A 
 
 = s 
 
 
 S 
 
 
 f( 
 
 
 = 8 
 
 
 = ' 
 
 
 F 
 
 COS 4 A 
 
 = 8 
 
 
 = 1 
 
 tun A A 
 
 _4 
 
 Dividing num. and den. by cot A 
 
V'i 
 
 — 01 — 
 
 ~ 3 cot it — tan i4 
 Sin i A = sin (3 A -\- A) = Bin 3 A COS A -{- cos 3 A sin A 
 
 (201) 
 
 Substituting for nin 3 A and cos 3 ^4 their ralues m 
 found in (1961 and (197), and reducing; 
 
 = 8 sin il cos" A — 4 sin ^4 cos ^4 
 
 = (4 sin ^ — 8 sin' A) ^l^— sin' A] 
 
 FOi 'owing a similar process gives 
 cos 4 i4 = 8 c 3* ^ — 8 cos» A-^\ 
 = 1 — 8sin'i4 4-88in*^ 
 
 tun i A = 
 
 4 tan A — 4 tan' A 
 
 1 — 6tan*^-f-tan*/< 
 Dividing num. and don oy tan A 
 4 — 4 tan' A 
 
 (202) 
 (203) 
 
 (204) 
 (205) 
 
 (206) 
 
 cot 4 .1 = 
 
 cot A — 6 tan j4 + tan" A 
 
 coi*A—6tot'A-^-l 
 4 cot' A -\- 5 cot A 
 
 Dividing num. and den. by cot A; 
 
 cot' A — 6 cot il + tan .(4 
 ~"4cot* A-i-li 
 
 (207) 
 (208) 
 
 (209) 
 
 £y following a similar process of successive substitutions, we 
 easily obtain the angular functions of 5, 6, 7... times i4. 
 
 But we may more briefly obtain the proper result for any mul- 
 tiple Angle, by means of the following 
 
 ,' "it 1 
 
 [ 
 
 
'' v4 
 
 —OS- 
 
 Genera/ Formulae for Multiple AngUi. 
 
 p"Mituting (w _ 1) 4 fbr one of the anglea in [Mb] and (156), 
 
 we obtain ^, , • a 
 
 sin r(m — l)i4 + -4] = ainim — l)il co8i4 +C01 («» — I) ^8in^ 
 
 Bin [im _ 1 ) .4 — il] = sin (m — 1) i4 cos .4 — co8 [m—\)A sin i4 
 
 Adding: 
 
 sin [(»» _ 1 ) /I + yl] + 8in [(7«-l ) ^ -^] = 2 sm (m-1 ) X cos yl 
 
 Whence : 
 
 Bin [im— 1 ) .4 + 4] = 2 sin (m-1 ) il COB .1 — sin [(wi— 1 ) i4 — X] 
 
 or 
 
 sin m ^ = 2 sin [m _ 1) /I cos /I — sin (w — 2) i4 l210) 
 
 Proceeding in a similar way with (157) and (158), we obtain 
 cos VI A = 2 cos (»i — 1) ^ cos A —cos {m — 2)A (211) 
 
 Substituting im - 1) yl for one of the angles in (159) and (161) gives 
 
 tan (wt — 1 ) yl-fton.4 
 tan m A = ^Zlta'nl^^i-l) A Xm A 
 
 cot m — 1) Aooi A — 1 
 
 cot HI A = 
 
 cot (HI— i) ^ + C0ti4 
 
 (212) 
 
 (2131 
 
 The formulse (210), (211), (212), and (213), are general and may 
 bo applied to the computation of any multiple angle, by making m 
 equal t<il, 2, 3, 4, 5, etc , and perfoiming successive substitu- 
 tions, as seen before. Thus, making m = 2, 3, 4, successively, we 
 have by (210) : 
 
 sin2^ = 2«in(2 — 1)^ cos ^ — sin (2 — 2) i4 
 = 2 sin i4.cos i4 = (179) 
 
 sin 3 4 = iJ sin (3 — 1) /I cos A — sin (»-- 2) A 
 
^ 
 
 -f.9- 
 
 = 2 will 2 A v.uH A — MuA 
 
 wliioli liuiuy nMliiciid Ity .sul)stitiitiuy for sin *-' A 
 its valuo ((179), givi's |19G). 
 
 sin 4 yl = 2 sin (4 — 1 )^ cos A — Hin (i — 2) A 
 = 2 sin 3 i4 cos /I — sin 2 A 
 
 wliiali Ijoiug reduced by the Ruccossivo sulwtitu- 
 tiou of .T A and of 2 A, gives (202). 
 And so on for the siuo of any higher multiple 
 of /I, and also for the other functions of any 
 multiple of ^, 
 
 AmjuJar Funefions of ^ A . 
 
 Substituting ^ A for A in (184) gives 
 cos j4 = 1 — 2 oin* A A, whence 
 _ 2 sin* ^ A=l~coii A; therefore, 
 //I— cos /I \ 
 
 sin ^ A 
 
 f 
 
 = m-V[l-ein»^3j 
 
 Substituting J ^ for A in (183) gives 
 cos ^ = 2 cos' ^A — 1; whence 
 2 cos* i} i4 = 1 -f- cos i4j therefore 
 
 »-i-7(i±l^) 
 
 =V(^ 
 
 [I — sin'^lj^ 
 
 (214) 
 
 (215) 
 
 f 
 
 r 
 
 (216) 
 
 (217) 
 
 ! 
 I t 
 
 i 
 
 ami 4 
 
 tan * yl = ~~ 
 
 cos ^ A 
 
 _ i n—co3A \ 
 
 V\l + cos^/ 
 
 ns) 
 
,*■• 
 
 i!' 
 
 
 
 -co— 
 
 Mulliplyin;^ num. and ilon. by ^ [1 — cos /I] or 
 ^[1 + C.11 A], ami reducins : 
 
 1 — con A 
 
 sin A 
 
 flin A 
 
 1 + ooa A 
 
 also 
 
 sin A 
 
 cot A -4 = -.--^^ 
 
 cos /t 
 
 cos 
 
 •(s^^')= 
 
 sec 
 
 tan A 
 
 sin 
 
 hA 
 
 lt\ -f-cos i1 V 
 
 ' "Y \T^ (OB /I / 
 
 Procoetling 03 Ixiforo ; 
 
 1 + cos A 
 sin A 
 
 sin A 
 
 1 — cos A 
 
 sin yl 
 cos .1 
 
 tan A 
 sec i4 — 1 
 
 sec 
 
 hA = 
 
 cos 
 
 >cos ^i / 
 
 (219) 
 
 (220) 
 
 (2211 
 
 (222) 
 
 (223) 
 
 (2241 
 
 1225) 
 
 
 
^, 
 
 -Gl— 
 
 cosec 
 
 
 _ n 2 8i!C A V 
 
 (227) 
 
 Formulas nxpresshm thit sum or the difference of two 
 Functions ineludinij one or two variahleH. 
 
 By a simple algebnuc artifice, we may take 
 
 h = ^[A-\-li]~}i (A —Ii\ whence 
 
 Sin A + mMB = sin [^ U + /i) + i (/I _ /ii] -|- mn [i [A + 7?) 
 
 — i(^ — Z/i] 
 = 8iniU4-iy) cosi U— Z/) + co8j (il + /i) 
 
 sill i (i4 — B\ 
 -f sin ^ U -f //( cos i (yl — /i) — cos i Ii4 + /i) 
 
 sin ^(.4 — /?) 
 = 2 sin J (i4 -f(fl) co=f i (/I - /y) (2281 
 
 Sin A — sin /i = sin l\^A -\- B] + ^\A — /ii] — sin,[^ (A + 7y) 
 ■-i(/l — 73)] 
 = sin i 1^1 + 7i| cos ^ (/I — 7^1 + cos ^ ul + U) 
 sin ^ 1/1 — Z?) 
 
 — sin i (^ + ^1 cos J (yl _ J3) + cos ^ (/J 4- 7?) 
 
 sin \[A—B) 
 = 2 cos i ( /I -f ^) sin ^ ;-4 — B\ (229) 
 
 Cos ^ -I- COB i? = cos [i (/i + J3) + i (^ — 7?)] + cos [i (/I + zy ) 
 _i(yl_7yi] 
 
 = cos i (^ + 7i) cos i (i4 — 7?) — sin J (.4 + 7? ) 
 sin J (yl — B\ 
 
 + COR i [A .+ 7f) cos ^ (^ — 7?, + sin A (/I + 7?) 
 
 
 ft» 
 
 i 
 
.t* 
 
 -OJ- 
 
 sin k iA — //i 
 T= L' COR K lA -f /;. ruM J, ,.-1 — // 
 
 rj:t<» 
 
 ,;oH /I .- COS 7/ = coH[hA 4- //( -f A (/I — /yij — cos [A i/i + yy) 
 — A(/i — y/i] 
 
 ~ cos i (/I 4- //) coa i 1/1 — //I — sin i (yl + /A 
 
 Min A {A — //) 
 — coH i ( yl + 7^1 coH A I/I — //) — Hill J {A -j- /ii 
 
 sill A (i4 — li) 
 
 Mill i ./I + //I sill A ul — //, 
 
 (2311 
 
 Aiu A-\- COH A — -j- ^ [m\\ a -\- cos A)''], which in I'vidont 
 — + v[ *•'"' ■^ + •2°^" ^ + 2 sin >! COH /I J 
 = 4- V[l + '''"2i4],l)y il79ianiUh(MM|imtioJi[a] 
 
 (2321 
 Ihit is -4-v'[14-'<in 2 /IJ 
 
 1 , for (ill viilufs of A wliosi' sin ami cos aro both 
 positive; 
 
 2. for values of A wIiosp sin nnd cos have unliki^ 
 signs, wium sin is groator and positivo, or 
 loss and negativo, f.itherwiso: 
 
 sin A — cos /I = -f- a/[('''° ^ — ""^ ■^'']' which is ovidout 
 = -f- -^/[sin' A -|- cos* A — 2 sin /I cos >1] 
 = + W —• «'° 2 A] (233) 
 
 that is = + ^[1 — sin 2 A] 
 
 1. for ail values of ^ whoen sin is positivo and 
 cos negative; 
 
 2. for values of A whose sin and cos have un- 
 like signs when sin is jjreator and positive, 
 or less and negative. Otherwise: 
 
 = —^[l—Bm2A\ 
 
 „ sin i4 , sin i? 
 
 tan A4-ta.n B= 7- + ;; — ^ 
 
 cos A cos a 
 
_cn— 
 
 n 
 
 coH [A \A -f- H) 
 »iu i (yl + Ji. 
 Hin J 1/1 -|- JJ\ 
 
 I (2311 
 
 H cviiloiit 
 
 [ COM A] 
 
 tlic ('i|imtion[a] 
 i232i 
 
 ml cos arc lioth 
 
 I cos have unlik<! 
 d poMitivo, or 
 
 in eviilcut 
 A cos A] 
 
 (2331 
 
 in is positive and 
 
 ind cos have un- 
 ter and positive, 
 
 rwise: 
 
 sin A cos // -f- CO" '^ si n JU 
 A CO* B 
 
 »lno = 
 
 Inn i4 — tan // = 
 
 uos 
 
 Hin {A 4- B) 
 
 COS A COH H 
 
 ' + 
 
 oot J9 4- oot A 
 
 cot 
 
 coti> 
 
 cot A cot yy 
 
 sin /I gin B 
 
 cos /I 
 
 COB fl 
 
 sin i4 COS ^ — cos il sin B 
 cos A cos /f 
 
 Bin (i4 — /i) 
 cos A cos £ 
 
 also = 
 
 cot A 
 
 cotB 
 
 cot B — cot A 
 cot A cot B 
 
 ^ „ cot A , cos /? 
 
 cos A Bin B-\- sin ^4 coa B 
 sin i4 sin Zf 
 
 si n {A -{■ B) 
 sin A sin if 
 
 also = - 
 
 tan A tan B 
 
 tan A + tanjB 
 tan j4 tan B 
 
 cot j4 — cot /i = 
 
 cos A cos i^ 
 
 sin B sin ii 
 
 (2341 
 
 (2351 
 
 (236) 
 
 (237) 
 
 (238) 
 
 (239) 
 
 I 
 
f^ 
 
 —64— 
 
 cos A siu B — sill A cos B 
 
 sin ^ ain £ 
 
 .in [B — A) 
 
 sin ^ sin B 
 
 (240) 
 
 alno •= - 
 
 1 
 tun A 
 
 tan /i 
 
 tan i? — tau A 
 
 tan ^ tan /I 
 
 i241) 
 
 cot A — 
 
 tan i4 -f cot fi 
 
 •4- 
 
 cos B 
 
 Hin 
 
 cos A sin J5 
 
 cos j4 cos li -}- sin il sin B 
 cos i4 sin B 
 
 COS (i4 — fij 
 
 also 
 
 cot A — tan i? 
 
 cos A 
 
 9in5 
 
 1 
 
 cot A 
 
 ' tan 5 
 
 cos A 
 
 sin B 
 
 sin -4 
 
 cos jB 
 
 cos A 
 
 cos /5 — 
 
 
 sin A C( 
 
 _ cos ;A4-A) 
 
 tan 5 + cot i4 
 tan B cot -4 
 
 sin A sin J3 
 
 i/ 
 
 sin A cos 5 
 
 (242) 
 
 (2431 
 
 (244) 
 
 sec 
 
 ^-f-B 
 
 sec B — Bci 
 
 also ~ 
 
 tan j4 
 
 cot B 
 
 cot fi — tan i4 
 cot B tan ^ 
 
 (245) 
 
 cosoc ^ -f- cq 
 
 tan Ar^oot A = 
 
 sin A I cos i4 
 COS A '^ sin A 
 
—65— 
 
 
 (240\ 
 
 ^ (241) 
 A 
 
 (242) 
 ^. (243^ 
 
 ^.A (245) 
 an A 
 
 _ Bin* A + (JOS' A __ I 
 
 sin A COS A ~ sin il coa il 
 
 cot A — tan i4 = 
 
 2 sin i4 cos A sin 2 il 
 
 008 A sin il 
 sin il COS il 
 
 cos* il — sin" il 
 sin il cos il 
 
 COS 2 il 2 cos 2 il 
 
 sin ilcos il sin 2 il 
 
 = 2 cot 2 il 
 
 sec A-\-6ecB= -, -- 
 
 cos il coaB 
 
 1 cos il -|- cos £ 
 
 COS il cos B 
 
 _ 2 COS ^ (il-f B) co8 j iA — B) 
 
 cos il COB B 
 
 soc B — sec il = 
 
 1 
 
 1 
 
 cos il — cos 5 
 
 COS B cos il cos il cos B 
 
 _ 2 sin i (il + B, sin i(il —B) 
 
 coscc A -f- cosec B 
 
 cos il cos B 
 
 1 sin il -{- sin B 
 
 sm .1 
 
 sin B 
 
 sin il sin B 
 
 2sin ^ (lil 4- B\ cos ^jA—B) 
 slJ il sin B 
 
 (246) 
 
 (247) 
 
 (248) 
 
 (249) 
 
 (250) 
 
 I 
 
it 
 
 I 
 
 m 
 
 -66— 
 
 cosec B 4" cos^c ^ = ^^ 
 2 cos ^ 
 
 1 sin A — sin B 
 
 cot iA + /fi 
 
 sin 4 sin A sin 5 
 
 J^ + g) sin k (^ — jgL 
 sin A sin 5 
 
 (2611 
 
 in(i4 + B) + 8m(4-B) = 
 
 sin 
 
 sin(il + 5) — 8in(i4 — 5) = 
 
 cos 
 
 COS 
 
 (A+B) + co8(il — 5) = 
 
 (^ _j_ 5) _ COS (il — /<) = 
 
 COS (A — B) — COS (il + 5) = 
 tan(^ + 5) + tJin|^-2?) = 
 
 
 
 sec {A-^/i) 
 
 (166) + (156) 
 2 sin il cos B 
 
 . (262) 
 
 
 (155) — 166) 
 2 cos A sin 5 
 
 (263) 
 
 sec {A + B) 
 
 ,167) + (158) 
 2 COB A cos £ 
 
 (264) 
 
 
 (167) — (158) 
 2 sin i4 sin B 
 
 <265) 
 
 COS(!C (^ + J 
 
 ,158) — (167) 
 2 sin i4 sin B 
 
 ,159) + (160) 
 
 (?66) 
 
 cosec (^ -|- , 
 
 2 tan il + 2 tan 4 tan fi* 
 
 (257) 
 
 
 tan(^ + IJ)-tan(/l--B) 
 
 cot'a + «) + cot(yl— -tf) 
 
 1— tan»il tan»i^ 
 
 : (159) - (160) 
 
 2 tan 5 + 2 t an* ^ tan .g 
 1 — lan*H tan* B 
 
 (161) t (162) 
 
 2 cot ^ + 2 cot A cot* -» 
 cotT/i -- cot* ^ 
 
 (268) 
 
 ,269) 
 
 Let VI be 
 vious tliiit 
 
 I228i (229' 
 sin iwi + 1 1 
 
-C7— . 
 
 cot i7l + /?i— coliyt — /ii= 1I6I1— (162) 
 
 _ — 2 cot /? — 2 cot A cot n 
 
 ~ cot'''"^ — 'cot» A 
 
 social + yil 4- sec {A — D\= il64l + il66i 
 
 2 soc A sGc B cosdc- A ensue''' li 
 
 (2601 
 
 V-J261, 
 
 = cosec' A cosec' /^ — sec' A sec" fl 
 sec |>1 + fil — ace (/I — 7ii = (164l — (166' 
 
 i sec' A sec ' i/ coaec A cosec if 
 cosec* yl cost'c/r' — secM.sec' 7^ 
 
 cosec 1^ -f- ^1 + <'0''"C 1-^ — 7^' = 1I681 f > 171)1 
 
 2 sec" ^ sec li cosec /I cosec' Ji 
 
 sec' j4 cosec'^ i^ — sec' H cosec" yl 
 
 cosec \A -\- Bi -- cosec lA — li\ = 1I68! — (170i 
 
 _ 2 sec il coaec B cosec' A sec' li 
 spc " ^4 cosec' 7? — - cosec' A sec' 7i' 
 
 263) 
 
 264 1 
 
 lj3t m be any positive number greater than unity; then il is ob- 
 vious that 
 
 ■\ m -\-\\-\-^ \iH — 1 1 =r m 
 h iw -f- li — h\m — ll =: 1 
 or (w -f 1 i + ()/( — 1 1 — 2 hi 
 
 (»i + 1 ) — [in — 1 1 = 2 ; whence, by 
 12281 (229', i230i, (231): 
 
 sin iw -{-\\A-\-6\vim — \\ A =2 siD[-Jr m -f 1 1 ^ -f. \{:n— 1 )A] 
 
 X cos ["i {m — W A — I [\\\ — \\a\ 
 '■■ i . > =2 sin m A cos A (26fK 
 
 t 
 
 \ 
 
 ■ ■! 
 
 4 
 
I ?> ; 
 
 -68— 
 
 sin (/n+ 1) 4— sinuft— 1 ) A =2 cos [^ (m+l) A -\-^(m—l) A"] 
 
 X. sin [J(m-fl) il — i (m— l)il] 
 
 = 2 cos m .^ sin i4 
 
 1 266) 
 
 co»{m-^\)A-^coa{m—h A = 2 C08[J {m + I) A+ ^{m —1) A] 
 
 X cos [^ (ni-f l)i4 — i(»« — 1)^] 
 
 = 2 COB 7n A cos A 
 
 (2671 
 
 cos (m -f l)A— cos (ffi — 1) ^ = - 2 8in[^ (»« + 1 U + i{m—l)A] 
 
 8in[i (»( + 11-4— itni — l)i4J 
 
 = — 2 sin «) A sin i4 
 
 1 268) 
 
 tan (m + DX + tan (m - 1) i* = ■ --- 
 
 sin l(m+ 1) il + (m— 1) .4] 
 
 cos (m + 1 1 .4 cos i»i — 1 ) il 
 
 by (2341 
 
 sin 2 m A 
 
 cos [m 
 
 + \\ A cos (»i — li A 
 
 i269i 
 
 sin 
 
 tan (Mi + l)i4— tan(w — Dil = - 
 
 [(7/. +1)4 — (w — 1) A]^ 
 
 cos |7« H- 1) A cos Iffi — 1)4 
 by (236) 
 
 sin 2 4 
 
 cos (»« + 1)4 Ci.t8 iJrt 1) A 
 
 ,(2701 
 
 sin [(»» + 1 )4 4 (w — 1) 4 ], 
 cot (TH 4- 1) ^ + cot iw - 1) ^ = ~8in (nt + 1) 4 sin (m- 1) 4 
 
 by (238) 
 
 cosec (m + 
 
 sin 2 m 4 
 
 sin(f» + 1)4 sin (m — 1)4 
 
 =ro <^"i 
 
' > I >l 
 
 ^(m— 1) A] 
 
 him-DA] 
 
 (266) 
 
 hil'n-l)4] 
 (267) 
 
 (268) 
 
 
 — i269i 
 
 m — h A 
 
 cos im — h A 
 
 . , (2701 
 
 . + \m—\\A], 
 sin (wi — ll A 
 
 1 ,071, 
 
 -69— 
 
 oot tm — ll i4— cot »i + I) A = -. — != TT'ii—- — ; vnr 
 
 am [tn + I) A sin (m — I) A 
 
 by (240) 
 
 siu 2 il 
 
 sin (m -f 1) i4 sin (»i— 1) A 
 
 i272) 
 
 , . cosfm+lM — »» — 1)^1. 
 
 tan (»i i-l)A+ cot m- 1) ^ = '■','■ y-T 
 
 cos [Tn + I) A sm (?« — li A 
 
 by (242) 
 
 cos 2 A 
 
 cot (»« + IM — tan'm — \] A = 
 
 co8( m-{-\) A sin (jw — 1) A 
 
 cos [(7?/. + I) ^ + (wi— 1) A} 
 
 (273) 
 
 sin (»M + 1) A cos (m — 1) A 
 
 by (244i 
 
 cos 2 III A 
 
 sin (m + 1 1 X cos (m — I) A 
 
 (274) 
 
 By applying the same method of substitution and reduction to 
 the formulse (248), (2191, (260), (261), we easily obtain: 
 
 sec {m 4 1 ) X + sec (m — 1 ) ^ = 
 sec (m — l)i4 — secfm + l)^ — - 
 C08ec(»i + \)A +■ cosec(w — l)^ = 
 cosecdn — l)il — coseolwi + \)A = 
 
 2 cos m il cos il 
 
 (m — M il 
 
 cos (TO + 1)^ cos (Ml — 1) A 
 
 (275) 
 2 sin m il sin A 
 cosiw-f liil cos( wi — I) A 
 (276) 
 2 sin m i4 cos il 
 sin [ni-\-}) A sin (m — 1) A 
 (277) 
 2 cos in A sin A 
 sin (m + 1) ^ sin m — 1) ^4 
 (278) 
 
 
 i.F 
 
 
Let A I"' 45°, 30'. or GO", in il'52i, i253i, i2')4i. and iL'5r>i: tlu^n 
 wi' obtain innuyiliattdy ; 
 
 sin (4B" + //I + ^in t45° — Ji\ = 2 sin 45° cos B 
 
 sin i45° + /;• — sin i45° — 7}i = 2 cos 45° sin Ji 
 
 = sin n V- '-80' 
 
 COS (45° + ]i\ + cos i45° —Bt = 2 cos 45° cos U 
 
 = cos n ^2 (2811 
 
 008 (45° + i^) — cos (45° — i)'| = — 2 sin 45" sin B 
 
 = 2sinyyv- '282i 
 
 sin (30° -\- B) + sin (30' —B) = 2 sin 30" cos /i 
 
 = cos /^ (283; 
 
 sin (30° + B\ — sin (30° — B\ = '1 cos 30° sin B 
 
 ::= sin y>' v3 (2841 
 
 cos (30° + B\ + (OS (30° — B\ = 2 cos 30' cos B 
 
 = cos B y/?, (2851 
 
 cos (30° -I- B\ — cos (30' _ yyi = — 2 sin 30' sin B 
 
 = — sin B (28oi 
 
 sin (60° ■\-B\-\- sin (60° — i^l = 2 sin 60° cos B 
 
 = COM B v'S i287i 
 
 sin (60° -\- B\ — sin (60° — H) = 2 cos 60' sin B 
 
 = sin B (288 1 
 
 cos 1 60° + B\ + cos (60° — yyi = 2 cos 60° co, 7V 
 
 = cos B (289) 
 
 cos (60° + i^i —cos (60° — B) = — 2 sin ()0° sin B 
 
 (290) 
 
 Formm 
 ein (^ 4". 
 
 cos (A-{-l 
 
 tan (-4 + y 
 
 = — 8in 
 
 B ^3 
 
—71— 
 
 265 1 : tlwn 
 
 (2791 
 
 li 
 
 B 
 
 1 280 1 
 
 i281i 
 
 (282) 
 
 (283: 
 
 (284) 
 
 ,28r)) 
 
 (286. 
 
 (287t 
 
 1 288) 
 l289, 
 (290) 
 
 Formulae expressing the Product of two Functiom including, 
 each, one or two Variables. 
 
 Bin (A + B) sin (A — B) = (165) x (156) 
 
 = (sin A cos B-{- cos A sin B) 
 X (sin A coa B — cos 4 sin B) 
 = sin" A cos' B — cos' A sin' B 
 = sin' A (I— sin' B) — (l — sin' A) 
 X sin' B 
 
 = sin' 4— sin' 5 (291) 
 
 or = (1 — cos' A) cos' B — cos' A 
 
 X (1 — cos'fi) 
 • = cos' 5 — cos' A (292) 
 
 cos (A 4- B) cos (A — 5) = (157) x (168) 
 
 = (cos A cos B — sin A sin B) 
 X (cos A coa B-\- sin A sin B) 
 = cos' A cos' B — sin' A sin' B 
 = cos' -4 (1 — sin' 5) — (1 — cos' il) 
 X sin' B 
 
 = cos' ^ — sin' 5 (293) 
 
 or = cos' fi — sin' i4 (294) 
 
 or = cos' A + cos' B — 1 (296) 
 
 tan (A + B) tan (A — B) = 
 
 also = 
 
 sin (A + B] bin A - 
 
 -B) 
 
 cos [A -{- B) cos [A - 
 
 -B) 
 
 sin' -4 — sin' J3 
 
 
 cos' A — sin'^ 
 
 
 cos' B — cos' A 
 
 
 cos' B — sin' A 
 
 
 tan' ^— tan' 5 
 
 
 1 — tan' zltan' i< 
 
 (296) 
 (297) 
 (298) 
 
 . ' i: 
 
 >^l 
 
—72— 
 
 ! « 
 
 cot -51 cot (A-B] = co8^(^ -f-^)coB(^-^) 
 
 sin (yl -\- H) sin (4 — J3) 
 
 __ cos* A — sin* ^ 
 ~ sin* A — sin* J? 
 
 cos* J9 — sin* i4 
 
 cos* ^ — cos* A 
 
 299) 
 (3001 
 
 cot A cot J? 
 
 cot»i4cot*fi— 1 
 
 .» = ae.,M.c. = ^— ^-i«, 
 
 8oc [A 4- J5) sec (A—B) = (164) x (166) 
 
 __ sec' A sec'^— cosec" ^1 coaoc' 5 
 cosec' ^ cosec' ^"^ec'I mo'^^ '^^^' 
 
 cosec (/l+i?l co-sec (.'1~J9) = .l^cM 8«;»^_cosecM co seo^Jg 
 
 sec ■ A cosec' // - nee' li cosec' -4 ' 
 
 In the preceding fonnnlae, write ^ (^ 4- B\, for A; and* (^ -/i, 
 for B, then since j^ (.4 ^ B)-\-^ [A — B) = A " 
 
 and i (^ + yy, = I (^ _ /?, = ^^ ,y^ j,t,t^i^ 
 
 •in .1 ain 5 = sin' ^[A-\-B) -sin' J (^ _£) 
 or = cos»^(^_Zfl — cos' J(^-f 5) 
 
 cos AcoiB= cos' ^ (.4 +£) — 8in' ^(A~B) 
 or = co8'i(^_5|_gin«|,^^^j 
 
 tan ^ tan i^ = ^^J^±±Z:^i^ ^ U^-Bl 
 . cos' i (^ + i^) _ Pin-.' ^ ^A—B) 
 
 or = co"'^ !^ — ^^— co^^ (A — B) 
 cos* ^(A—B) — sin* i (.4 — ^) 
 
 (304) 
 (305) 
 
 (306) 
 
 (307) 
 
 (308) 
 (309) 
 
-73— 
 
 cot A COl JJ = ■ , 1 , . I — 57" — . . . , . 5: 
 
 Bin* ^{A-\-B) — sin* J (A — J?) 
 
 (SIO) 
 
 
 cos' ^ (/I — B) — ain' ^ (i4 -{- g) 
 co8» i{A — B) — cos' i (/I -f fl) 
 
 (311) 
 
 Fomtulas expressing the quotient of Functiont including, 
 
 eachy one or two Variables. 
 
 sin {A-\-B) _ sin A cos 5+ cos ^ sin B 
 sin \A — B) sin A coh B — ooa A sin B. 
 
 Dividing nnm. nnd den. by cos A cos £ or sin A sin jR, and re- 
 ducing 
 
 k%. 
 
 tan il 4 tan B 
 tan A — tan B 
 
 ,312) 
 
 cot B + oot A 
 cot B — cot A 
 
 l313) 
 
 Also dividing num. and den. by sin A cos B, or 
 cos A din B -h. 
 
 1 + tan J5 cot A 
 1 — tan B cot A 
 
 (SUi 
 
 tan A cot Jg -f 1 
 
 tan A cot B — 1 
 
 (315) 
 
r 
 
 # 
 
 
 
 
 
 _74_ 
 
 
 
 
 • •• 
 
 ...,^ 
 
 , 
 
 cos 
 
 CCI8 
 
 
 — B\ 
 
 i!08 A 
 
 COH li 
 
 — Bin A sin // 
 + siu it 8in B 
 
 
 
 
 
 
 
 coa A 
 
 CON B 
 
 
 
 
 
 J)ividi 
 
 ng iiuiii. and dim. hy (;o« 
 
 A 
 
 HJn 
 
 «, 
 
 or 
 
 
 1 
 
 
 
 
 8in 
 
 A COH 
 
 B, and reducing: 
 
 
 
 
 
 
 cot yi — tiiQ A 
 
 cot /y + tau A 
 
 (316) 
 
 .,... 
 
 cot A — tau B 
 cot i4 + tan yy 
 
 (317) 
 
 Also dividing num. and don. by cos A cos B, 
 or Hin A sin B, and reducing: 
 
 1 — tan yl tan B 
 i + tAnA~t&n~B 
 
 (3181 
 
 cot i4 cot B — 1 
 cot >1 cot jB + 1 
 
 By following a similar process, wo obtain ciwily 
 
 tan {A 4- B ] _ sin A cos B-^ cos A sin B 
 cos {A-\-B\, C08 A cos .ft — sin X oin .S 
 
 - .^ ~f- cot A tan yi 
 cot A — tan B 
 
 _ tanyl cot y y-f 1 
 cot5— t.nn^ 
 
 (319) 
 
 (320) 
 
 (321) 
 
 ^inA 
 
 Bin {A — I 
 
 siu.^ c 
 
.).} — 
 
 si 
 
 tan i4 + t'U J I 
 
 1 — tan yl tan B 
 cot J} -\- cot A 
 
 cot A cot li — 1 
 
 (32'J 
 
 (323) 
 
 1^ 
 
 sin {A — B\ _ 1 — co\^A tiin H 
 COR (i4 — B) cot i4 -f t)in7/ 
 
 _ tan i4 cot J9 — 1 
 
 cot B -}- tan A 
 
 tin i4— tan /? 
 
 1 -4- tan A tan i3 
 
 cot B — cot i4 
 cot A cot B -\- 1 
 
 (321 1 
 
 (3251 
 
 1 320, 
 
 (327) 
 
 [ 
 
 Bin [A + B\ _ I + cot ^ tan 5 
 cos [A — B) cot A + tan B 
 
 _ tan A cot /i + 1 
 ~~ cot i? + tan ^ 
 
 __ i&n A-\-\an B 
 ~ 1 + tan AtanZJ 
 
 __ cot 5 + cot il 
 cot A cot B-\- 1 
 
 Bin (i4 -}- -fl) _ sin ^ cos B-^-aoa A sin B 
 sin A toiB~ em A cos iP 
 
 = 1 + cot il tan £ , 
 
 sin [A — B) sin il cos ^ — cos A sin B 
 
 (328) 
 
 (329) 
 
 (330i 
 
 (331) 
 
 (332) 
 
 sin A cos B 
 
 sin ^ cos B 
 
-76- 
 
 • I 
 
 
 = 1 — cot i4 tan B 
 
 cos {A-{- B) __ cos A cos B — sin i4 
 
 (333) 
 
 sin 
 
 B 
 
 sin A sin B 
 
 sin i4 sin B 
 
 = cot 4 cot B — 1 
 
 cos (A — B] __ cos A cos ZJ-4- sin A sin B 
 sin A aiu B ~ sin i4 sin B 
 
 = cot i4 cot 5 -+• ^ 
 
 cos {A -{*B) _ cos A cos ^ — sin il sin fi 
 cos i4 cos B~' cos i4 cos B 
 
 = 1 — tan i4 tan ^ 
 
 cos (A — B) _ cos A COS B-\-ti\n A sin J9 
 cos i4 cos £> ~ cos i4 cos J9 
 
 = 1 4- tan i4 tan ^ 
 See also (234), (236), (238), (240), (242), and (244). 
 
 BinA + sin B __ 2 sin ^(A + B) cobHA—B \ 
 BiaA — amB~ 2 cos J iA+B) sin ^[A —B) 
 
 = tan J (il + 5) «ot J (il — fl) 
 
 - tani(it-f ffl I 
 
 ~tani(il — •*) 
 
 coaA+coaB __ 2 cos j^ {A + glcos ^ (it — B) 
 coaA—coaB ~ -2fm^^i4 7f<i]>) Mn J iA— JS) 
 
 (334) 
 
 C0« i< -f (!0( 
 
 
 sin A — sin 
 
 
 cos A -I- cos 
 
 (33S) 
 
 sin A + sin 
 
 cos A — COH 
 
 
 8in.i4 — sin 
 
 
 COH A — cos 
 
 (336) 
 
 sin {A + B 
 
 
 siii {A + ZT 
 
 
 cos (.4 4-^1 
 
 (337) 
 
 cos (i4 + //, 
 
 
 sin (-4 -f /ii 
 
 
 oo»{A-}-B) 
 
 
 sin (i4 -f 5) 
 
 
 <»a(A-^B) 
 
 (338) 
 
 siiHA-^B) 
 
 
 «w (i4 4- 5) 
 
 (389) 
 
 
 = — coti(i4-f £)coti(i4 — £) 
 
 (340) 
 
\ 
 
 — < I- 
 
 _ cot J (/I -f /?) 
 
 tan i (/< — /y 
 
 «in A fain /i _ (228) _ 
 
 cofli4 
 
 -\- 
 
 COS 
 
 B 
 
 sin A 
 
 — 
 
 ain 
 
 B 
 
 COB A 
 
 + 
 
 cos 
 
 B 
 
 siDil-f 
 
 Bin 
 
 B 
 
 coa il 
 
 — 
 
 COH 
 
 n 
 
 tin. A 
 
 — 
 
 sin 
 
 B 
 
 (230) 
 
 (22fl) 
 (230) 
 
 - tun i i/l -}- //) 
 
 = t«u i (.4--//) 
 
 _ (2281 _ 
 
 (2311 
 
 (2291 
 
 cofi^ —cos/? ~ (231) 
 
 cot i 1/1 -, ^) 
 
 — cot J :/l -f. JJ) 
 
 ain {A + B -^- ain l^ — //) (352) 
 
 sill 
 
 i/i + zy,- 
 
 sin ;i4 • 
 
 -B~ 
 
 cos 
 
 (^ + ^1 + 
 
 cosiilrf jOi 
 
 cos 
 
 [A + Bf- 
 
 COS 1^ - 
 
 -B) 
 
 sin 
 
 {A-]-B) + 
 
 sin tA ■ 
 
 — B\ 
 
 00* 
 
 M + fi) + 
 
 COS f4- 
 
 -B) 
 
 sin 
 
 1^ + 5)4- 
 
 ■in (^ ■ 
 
 -B) 
 
 OM 
 
 {A+B)- 
 
 cifa^il- 
 
 -B) 
 
 sill 
 
 U+B)^ 
 
 sin (i4 - 
 
 -B) 
 
 ^^ = ion. A cot B 
 
 <2_M) 
 (255) 
 
 = — ' cot /I cot // 
 
 _ |262) __ 
 
 tan A 
 
 (262) 
 (2£9) 
 
 . ^ (263) 
 
 cos (k 4- J?) — cos (^ -, B) (250 
 
 = cot B 
 
 »«nty4 
 
 Deductions. 
 
 By (183) and (184), ire have 
 
 2cofl*4s;i4.eD8 2/&l 
 2 sin* ii = 1 — cos 2 il 
 
 (841) 
 
 (342) 
 (343) 
 (344) 
 '345) 
 (34C) 
 (347) 
 (348) 
 (349) 
 
 (sno) 
 
 (381) 
 1352) 
 
 
 !■' ■ 
 
ff^ 
 
 - 7t<- 
 
 
 Hence 
 
 sin' A 
 
 coa* A = 
 
 1 — cos 2 A 
 
 1 4- COS 2 i4 
 
 l353i 
 
 (354; 
 
 Also 
 
 Whence 
 
 .^n4=7(i-T^-^) 
 
 (3551 
 
 co.X=V('±^^4) 
 
 (3561 
 
 2smH^ = l-.cos4 
 
 (357) 
 
 2 cos* J ^ = 1 + cos ii . 
 
 (358) 
 
 sin* i il = 
 
 cos* ^ A = 
 
 1 — cos il 
 
 1 + cos >1 
 2 
 
 Bin 
 
 cos 
 
 i-7(^^) 
 
 2 sin il = ^^[2 — 2 cos 2 ^4] 
 2 cos 4 = V[2 + 2 COB il] 
 
 2 sin J il = ^[2~ t cos 4] 
 2 cos ^ A = <^[2 4- 2 cos A'] 
 
 tan' i4 = 
 
 1 — cos 2 i4 
 1 + oot 2 i4 
 
 (359i 
 (360) 
 (361) 
 
 M = 7(i±|?l-^-) 1362. 
 
 (363) 
 (3641 
 
 • (365) 
 (366) 
 
 (367) 
 
 la (179)8.1 
 
 Henco 
 
 '•vnenco 
 
 sm 
 
 sin 
 
 % (196) 
 %(179tnn 
 
 f.c. 
 
■7'3— 
 
 cot* A = l±^l± 
 1 — cos 2 /I 
 
 tan 4= /(i-°°«2^V 
 
 cot- A= ML±^2^_v, 
 V \1— cos 2,4 j 
 
 tan* * ^ = i--co8 ^ 
 1 + cos A 
 
 1 COM A 
 
 In (179) subititutmg ^ A for A gives 
 
 sin ^ = 2 sin ^ ^, ..qs ^ ^ 
 Henco sin (^ + ^, = 2 sin ^ ,X + ^, cos 4 ;^ + //, 
 
 vvnenco 
 
 sin ^jin5^ _ cos i {A ~-B) 
 
 ain (A -f li] 
 
 cos i (^ 4- ^1 
 
 (368) 
 (369) 
 
 (370i 
 (371 1 
 (372, 
 
 (373, 
 
 (37 ii 
 
 (375i 
 
 ** 
 y 
 
 1 1 
 
 I i 
 
 J 
 
 l!y (179) nnd equation [alpaffe 13, t\-n have 
 
 sin 2 i4 = - ^ ^'° 4 ""^ -^ 
 co&» ^ ^-sin* ^ 
 
 I>ividingn„m.aml.,Io„. byco,«^ 
 
. '■:■■' -VK:>:.:^i^ir^x^'ai 
 
 -j'i 
 
 —80- 
 
 ! t 
 
 whence 
 
 2tanil 
 1+tanM 
 
 - 
 
 (376) 
 
 ■ cos(/4-f2? 
 
 . . 2 tan * ^ 
 
 sin A = -= 
 
 1 f tan* J A 
 
 
 (3771 
 
 1 
 
 Dividing num. and den. 
 
 hjtasxi A 
 
 
 1 tiinf^-f-j5. 
 
 2 
 
 
 lOTOx 
 
 1 
 
 cot J i4 -}- tan ^ il 
 tan* >1 — tan* B = (tan il + tan B) (tan i4 — tan B) 
 
 = (234) X (236) 
 
 _ sin {A 4- B) ein (.4 — B] 
 cos* ^ COS* /y 
 
 cot' .4 — cot? J? = (cot A+ cot B] (cot .4 — cot B) 
 
 = (238) X (240) 
 
 _ sin M 4-5) sin (^ — .41 
 ~ sin* A sin* B 
 
 — sin (.4 4- B) sin 1.4 — B]\ 
 
 (3791 
 
 (3801 
 (381) 
 
 sin* A ain' 5 
 
 Functions involving three Amjlea. 
 
 Let il be any 'hree angles; then sin (A-\- B-\-C) and 
 cos [A-^B-\- C\ may easily bo expanded by making use of the 
 formula (156) and(167) thus, 
 
 sin [A-\-B-\-C) = sin [A + Jf?) cos C+ cos [A +B) sin B 
 =8in A cos 5 cos 6' -f sin j9 ooa .4 cos C 
 -\- sin C cos i4 cos i? — sin .4 sin B sin C i382l 
 
 eot(^ -f n 
 
 It 18 manifl 
 tions of any i 
 tions of the 
 
 We have a| 
 sin (A ■\. B 
 
 + sin yA 
 = sin {{B + j 
 
 + 8in[.4 
 = 2 sin (B 
 
 JS* 
 
— «1- 
 
 |376) 
 
 1377) 
 
 (3791 
 
 0\ and 
 
 ing use of tlie 
 
 JfB) sin B 
 A cos C 
 n/JsinC «382i 
 
 cos ( A 4- iJ + C) = cos [A -f- i*) cos G— sin (A+ 5) flin C 
 = cos -4 cos B cos C — cos A sin i? sin C 
 — cos B ein ^ sin C — cos C sin A sin 5 (383 
 
 tan(yH-5 + C') = 
 
 sin (A -\-B-^C) 
 cos 1^1 4- i? + C) 
 
 eot(yl + i.' C' 
 
 Expanding num. and den. and dividing both by 
 cos A cos B cos (7, 
 
 tan A + tan i; + tan C — tan ^ lan i5 '.^an C7 
 1 — tim ^ tan 5 - - tan A tan C — tan ii tan C 
 
 (384) 
 cos [A + B + 0^ 
 sin lyl + y3 + C 
 
 Expanding and dividing by sin il sin B sin f?, 
 or by cos A cos Zf cos C : 
 
 cot ^ cot 5 cot C — cnt A — cot ii — cot G 
 
 or = - 
 
 cot B cot C + 
 1 — tan 5 tan C 
 
 cot A cot V + cot .4 cot B — 1 
 
 (385) 
 — tan il tan G — tan A tan B 
 
 tan /I + tan i* + tan < ' tan .4 tan fi tan G 
 
 (386) 
 
 It 18 manifest that in the same way, expressions for the Func- 
 tions of any compound Angle may bo found in terms of the func- 
 tions of tlie component angles. 
 
 We have also, by applying successively (262) and (155): 
 sin (4 + 5 + C) -(- sin [B -f a—A\ ''"n (A ^^ G — B) 
 
 + axD.^A + B — C) 
 = 8in[(B + C) -f .4] -I- sin l(B + G] —.4] -f sin \_A — {B—C)] 
 
 -♦-8in[il -f (5— Cl] 
 = Ssin [B + C] cos A ■{■ 2 sifi A cos [B—G) 
 
 '4\ 
 
—82- 
 
 = 2 sin A 'in /> sin ('4-2 sin A cos B cos C 
 
 -f- 2 tiiu /; cos A (;os C'+ 2 sin (' cos 4 cos i? (387) 
 
 I5y ajvplyinp; successively |25-li (157): 
 
 COS {A 4- B-\- G) + cos [B-\- G—A)-\- cos (-4 + C~ B] 
 
 + cos {A 4- 2? — r-i 
 
 = cos [|/J+'';i + yl]+co9[iiy+ G'l— ^] +cos[^ — 'i;~C'i 
 + cos[^ + i/? — Ci] 
 
 — 2 cos iB 4- G\ cos yl -(- 2 cos (7? — (h cos .4 
 = 2 cos ^1 2 cos B cos C 
 
 C08 i4 cos B 
 
 =z 4 cos A cos /^ cos C 
 
 1 388 1 
 
 [A 4- Zyi sin (2^4- '*' = «in i^ 4" -/^' «iii -^ f'os C 
 4- sini/l + /y) cos B sin C 
 
 = sin [A 4- ^1 sin B cos C 4- cos A sin Z? 
 cos /? sin C'4- sin A sin C cos' H 
 Suhstituting for cos' /iin the liist torni 
 its eciuijl 1 — siu'' B, and factoring: 
 
 = sin B [sin {A-\- B\ cos C+sin C 
 X (cos A cos ^ — sin A sin i?l] 
 4- sin A sin C 
 
 sin A + sin /J 
 
 cos A 4- cos i 
 
 = sin i4 sin C4- sin B [sin {A 4- 77) 
 X cos C-\- sin Ccos (yl +/il] 
 
 = sin A sin C4- sin B sm i A -\- B -\- C) 
 
 (389) 
 
 The student will have no difficulty in deducing the followiDg : 
 
tA^i^ 
 
 -83- 
 
 sin A+Ji f C'j 
 cos A cos B COS 6' 
 
 sin |yl + i^ + C'l 
 
 = tan ^ + tan i^ + tan C— tan 4 tan 5 tan C 
 
 {3901 
 
 mn 
 
 /l"S7ry7^f ' " '-'''^ '■ '■'^^ ^ + «>t /f cot C7 + cot 4 cot ;?_ 1 
 
 (391) 
 
 
 cos {A + B + C\ 
 
 ^A^B^^ = *'°* ^ '^^^ ^ '='>* ^- «>* ^ -cot /? -cot C 
 
 cos {A-\- B + Ci 
 cos ^ cos B CO 5 C' 
 
 — 1 — tan /; tan C— tau A tan C — tan A 
 
 (392, 
 
 t-anJ9 
 
 1393) 
 
 sin A sin (/>'— C'l + sin B sin |C— yt' + sin ( sin U _ ^j =_ 
 
 (394) 
 COB A sin IB— C) + cos 5 sin tC—A] + cos Cain iA--B) = 
 
 (395) 
 
 sin A + sinB + sin C = 4 sin ^ ii4 + ^) sin ^ (A + 
 
 C] 
 
 cos 
 
 sm 
 
 cos 
 
 sin i (i? + C) + sin (^ -f J? -f C)(396) 
 
 A+ cos5-f cos C=4co8 J(A + ^ico8j(^ + C, 
 
 cosi(/i-fC)_co8M + j5-f-C) (397) 
 
 {A — B) + sin{B—C,-i-ein(C—A) = — 4 s.n^ (.< — ^) 
 
 sin 
 
 he followina : 
 
 i(B~C}Bmi{C—A) ^398) 
 
 2 ^ 4- cos 2 ^ -f cos 2 C-f cos 2 (.4 -f i?-f C, 
 
 = 4 cos (^ + B) cos (^-f C; cos 1/1 -f C) (399) 
 
 { 
 
_84_ 
 
 Particular Case of A -{-11+0= 180° 
 Let A + B -\- C = IS0° OT he the angles of any triangle, then 
 
 8inU + ^+C) = 8iul80° = 
 cos (A + B+C) = 180" = — I 
 
 (400) 
 (401) 
 
 Kin 
 
 ^4- sin 7? 4- sin Cr= 2 sin i (A + B) cos ^ {A - 5) + sin C 
 
 But sin C = 2 sin i C cos ^ ^, 
 
 sin * C = cos ^[A-\- B), by (8i) 
 co3icr^8in*(^+£),by(83i 
 
 Therefore 
 
 sm 
 
 + 8inZJ+.sin C = 2am ^^A+B) [cos^U — /?) 
 
 -I- cos (^ + ^] 
 
 = 2 cos * a [2 cos i ^00? i B 
 = 4 cos ^ ^ cos A B cos i C 
 
 1 402! 
 
 CO? /I -f COB ^ 4- COS C t= 2 COS J U + 5) COS i (4 — 5) + COB G 
 
 = 2 8ini CcoB^iA — B) + 1— 2sin»iC 
 
 = 1 +. 2 sin \ C[cos \ lA—B) — sin i 0] 
 
 — I + 2 sin ^ C [cos hyA — B 
 
 — cos [A + 7^ 
 
 = 1 + 4 sin ^ il sin i 5 sin ^ 6' i403) 
 
 «m 
 
 'I A -\- >iin 2 i^ + sin 2 r- =r 2 sin (-44- B) cos (^ — 
 
 B\ 
 
 4- 2 sin C COB C 
 = 2 sin C COS (4 — J5) — 2 sin Ccos 
 
 (A+B) 
 = 2 sin C [cos (^ — .B) — cos (^44- J?)] 
 
-85— 
 
 = i Bin A Bin B sin C 
 
 1404) 
 
 sin (A +B) Bin (fi+ C)= «» (180° — O ein (180° — A 
 
 = sin A sin C (405) 
 
 l\ 
 
 PI 
 
 Bin (A-\-B+ C)4-8in M + 5— C)-f sin \A + 6'— Z?) 
 
 4- sill (^+ C—A) 
 = sin ISO"" + sin (180° — 2 Cl -f sin 
 
 (180° 2i;)8in + (180°— 2i4) 
 = sin 2 ^ -+- sin 2 ^ -f sin 2 C 
 
 = 4 sin ^ sin B sin C 
 
 (406) 
 
 Since 
 
 sin' i A-{- sin* i -B + sin i C+ 2 sin J ^ sin i J9 sin i (7 
 
 = 1 — (cos* iA— sin* J B) + sin* J C 
 
 -fsin ^ C (cos i {A+B) —cos (/I — ^) 
 
 =.-. 1 — cos i (^ + B] cm ^{A — B) 
 
 + sin* i C-f sin ^ C [cos J (A — fi) 
 
 — sin I ] 
 
 = 1 — sin J C cos ^ (^ - /?) + sin* ^ C 
 — sin J C'-f- sill J 6' cos A (i4 — /ii 
 
 = 1 
 
 'i'herofore 
 
 Bin" ^A -f sin*^ B + sin' K' = 1 — 2 sin U sin I B sin ^0(407) 
 
 
 '4\ 
 
 Since cos' A -f cos' Z? 4- cos' C + 2 eos /I cos B coa ^ ' 
 
 = 1 + (COS* A — sin' B) + cos' + cos C [cos ( 4 + i^) 
 
 + coa [A — Bi} 
 = 1 4- cos (i4 4- B\ cos [A — B) -i- eob' C + cos C [— oos (7 
 
 4- cos [A — jB] 
 = 1 — coa a cos (4 —if) 4- cob' C — cos* C+ cos (T 
 
 cos 1/1 — Bj 
 

 
 - iSd — 
 
 rr. 1 
 
 Tlun'ffoio 
 
 cos' A -f cos' Ji -f- cos' C -- 1 — 2 cos A cos B cos C(408) 
 
 Since tiui lyl + /i-f 6', = tnn ISO" = 0, tlion by (384) we 
 
 obtain 
 
 tan A -f tan JU + tin C — t&rx A t&n B tan O =■. 
 
 Vrhence 
 
 tan i4 + tan ^-f tan C= tan il tnn /i tan ^(409) 
 Thp student will easily deduce the following: 
 
 cos ^ i4 -f COS ^ 54 cos i 6'= 4 cos ^ {A + B] 
 
 cos J iZJ+ C) coa \[A-\- a, (410) 
 
 tan ^ i4 t«n i JB + tan i J? tan J C+ tan ^ ^ tan i C = 1 
 
 (411) 
 
 cot ^ + cot ii + cot 6' = cot A cot iJ cot C 
 
 -{-cosec^ cosoc B cosec C (412) 
 
 cot Jil + cotJ 5 4- cot ^ C- cot ^^ cot 4 Z; cot ^C 1413) 
 tan il + tan 5 + tan C _ tan J /I tan J i? <.an h C 
 
 (sin A sin B sin 6V 
 
 2 cos ^ cos yy cos V. 
 
 (414) 
 
 tan /I , tan i? tan 2? tan C tan A tan 
 
 + \. 4. + J. 
 
 tan B tan yl tan C tan B tiin C tan A 
 
 = sec ^ sec B sec (7 — 2 (415 ) 
 Pariicnhir casi' of A + B + C 
 
n tan C |409) 
 
 ;oscc C (412) 
 
 —87- 
 
 siu (yl + /i 4- C) = Hin 90° = 1 
 COS [A + B-\-C] = C03 90° = 
 
 (116) 
 
 (417) 
 
 sin 2A-\- sm-\-B + sin 2 (7 = 2 sin U4 + B) cos iA — B +sin 2 
 
 = 2 cosC [cos A — />') + cos \A+B)'\ 
 = 2 cos C [2 cos A cos ZJ] 
 = 4 cos^ cos // cos C (418) 
 
 Since tan |/1 -f Z? + C) = tan 90'' = oo, tliun by (384) wo obtain 
 tan A tan B + tan 5 tan C+ tan A tan C — 1 = 
 
 Whence 
 
 tan A tan B -f tan 7? tan C + tan A tan C = 1 (419) 
 
 Since cot {A -\- B -{- C, = cot 90° = O.thon by i385l 
 
 cot A cot jB cot C — cot A — cot B — cot C = 
 Wlience 
 
 cot /I + cot y? 4- cot a = cot A cot B cot C (420, 
 
 Since sin {A-\- B-\- C) = 1, the formula, i382i ))ocomes, after trans- 
 posing the last term: 
 
 sin A cos B cos C+ sin b cos A cos C + sin C cos A cos 7J 
 
 = sin A sin ii sin V + 1 
 
 Dividing botli num. and don. by cos A cos B cos gives 
 
 tan A + tan B -{- tan C = tan A tim B tan C 
 
 4- sec A sec Zi sec C (421) 
 
 
 ,i*^ 
 
 I 
 
 '-< 
 
1 1' 
 
 r 
 
 -an— 
 
 A/ijilicalion of the precediinj Furmulv to the 
 Jledu'tion of Trigonuinetrical Expremona, 
 
 Hih-'ito wn h;iv(), {i^ttablisliod tho fnu(l!imontnl f'ormnln' of Ana- 
 lytical Trigonometry, and disrived from thtiu n largo niimlicr of hc- 
 condary ones which may bo of frequent \\m of tho reduction of Trig- 
 onometric exproHsions from a complex form to a more simple one, 
 or to any such one as ia required to fulfil certain conditions. Tho 
 number of these foimultc could bo increased beyond limit; but the 
 student is now sufficiently prepared or tho elementary ajjplication 
 of Trigonometrical values to the othor blanches of scienc to which 
 tliey are connected. Wo shall close tlie present Chapter in giving 
 a few examples of reduction, indicating the methods to be used 
 in order to olloct tho proposed reduction in tlie first, and leaving 
 the others to the sagacity of the student, 
 
 sin A -j- sin iA -f sin 6A 
 1. Reduce Ihe expression -^-^_ -_^-___^--__- to 
 
 a single function? 
 
 ■ The proposed expression may be written 
 
 sin 3i4 + '«•" ^-^ -|- sin i4i 
 cos 3j4 -|- icoH .^A -\- cos A\ ■ 
 
 sin 3/14-2 sin 6A cos 2A 
 cos 3i4 + - <^os 3i4 cos 2/1 
 
 ,by .28; (fe ,233l 
 
 Factoring: 
 
 _ sin ZA (1 -f 2 cos 'lA] 
 cos iA (1 -|-coa 2A) 
 
 sin iA ^ , . 
 
 —^-r = tan oA. Ann. 
 
 cos oA 
 
 2. Reduce cos^il — em* A to a single function ? 
 
= cohM — nifi* A 
 = iios 2A. Ann. 
 
 ^. llt'duco Iff! A 4- '"" ■^ ^" " Mn^U' ruuution. 
 
 . 1 -f- "in 'I 
 
 sec A + tttu A — ~ — - 
 
 coH A 
 
 _ l_rus |<)()° + /1, 
 ■~ Hill i!»(P 4-/1 i 
 
 = tan (45° 4-^ .-li. Auk. 
 tnn I A-^ cot A A 
 
 i. TiimIuco 
 
 singlp function. 
 
 to IV 
 
 2 ~ - Vos i/1 "^ .sin \ Af 
 
 mi' \ 4" *-*"•■*' i ■' 
 2 .sin A .1 cm .'. /t 
 
 . »l 
 
 I t 
 
 
 '^ 
 
 1 
 
 = coHdc A. Ans. 
 
 sin i4 
 
 .">. U(i(lucc 1 4- tau ''I tun \ A ;o n ain;,'li' riuiotion 
 
 sin A 1 — co.s A 
 
 I -f- tan A tf»n i ^1 - 1 + 
 = 1 + 
 
 COS A sin i4 
 
 1 — (io.s A 
 cos /I 
 
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4^0 
 

 — yo— 
 
 ' \C0S /I 
 
 1) 
 
 \ , 
 
 - 1 4- iSL'C A — l\ 
 
 ■— SPfi A. Ali.i:. 
 
 G. Et'duot' «in i.30= -\- A]-\- sin liiO" _/l i to a single function. 
 
 ^/is. cos A 
 
 7. Reduce siu (46"^ -f- ^ i siii i45'^ — .4 1 to a single function. 
 
 Anf> i COH 2j4 
 
 8. Rcduco 4 sin A sin (GO°-f-^) sin 60° — Ai to a singlo func- 
 tion. Ans. .sin 3^4 
 
 9. Reduce sin A cos i/4 + III — cos A sin lA — Ji\ to an expres- 
 sion wliich can be submitted to loj^arithuiic eomiiutation. 
 
 Ans. cos 2A sin Z?. 
 
 , , ,. , sin ^ A + sin tA . , . 
 
 10. heduce ., , i -; - to ii sni'de funolioTi. 
 
 cos ^ A -^ cos J i4 
 
 Ann. tan yl. 
 
 1 1 . Reduce cos' ^ A {\ * >ec \ A ■}- tan ^ yl 1 1 1 — sec ^A+ tan i ^4 1 
 to a ein''le function. Aus. sin A. 
 
 ,, , "Jsui A — sin 2i4 . . ,. 
 
 12. Reduce - to suiL'le luuctiou 
 
 2sin \ -f- sin 2.4 
 
 Aii.i. taa' ^ A. 
 
 13. Reduce 'cos .4 -f cos /<'' + .sin .4 -f- sin Ih' to a sinp;l(j func- 
 tion- yl ".s- 4 cos* -^ 1 ,4 — Ji I 
 
 1 — iau* 4.5^— yli . , ,. ,. 
 
 14. Kuduce •, -■ ,,, - to a smj^'le lunrtion. 
 1 +tan- 1 45" — A) 
 
 An,'<. sin 2^1 
 
frlc function, 
 cos A 
 
 function, 
 i cos 1A 
 
 ) ii single fiinc- 
 8in 3i4 
 
 fl to an exprps- 
 
 ition. 
 
 . coH 2i4 sin B. 
 
 311. 
 
 tfin A. 
 
 Q,hA + tan I i4 ) 
 sin A. 
 
 3 a 8inj,'](! fiiuc- 
 , 4 cos* A \A — B\ 
 
 sin 'lA 
 
 -Oi- 
 
 ls. Rudues sin %A cosoc yt — co.s 3.4 eec A to a numerical va!;n. 
 
 Am. 2. 
 
 « 
 
 1 .' -u 1 COS j4 — cos 3^ 
 
 16. Ecduce ^.^^ .^^^ _ ^.^ ^ to a .single function. 
 
 17. fieduce 
 
 cos lA — C0.S 4^ 
 
 Am. tan 'lA. 
 '^n'W-^I^n-lA "-""-"'"gl'' function. 
 
 An». tan 3.,4. 
 
 18. Jieduce cos' U - /fi -f co8» /i— 2cos i^ _ J5) jos ^ cos li to 
 * «ngle function. ^^ g,^. ^ 
 
 19. Keduce sin' (^ _i?) -f- sin' ^4- 28in ^A~B\ sin i? cos A to 
 single function. ^,,,,. ,j„. ^ 
 
 20. Ruhice iicos" A sin SWi + ^.^n* ^ cos 3^. to a iingle func 
 ''""■ ^na. ;j- sin 4yl. 
 
 21. Iteduce^".i*±i^":'^ + «''"'2«i-ll^ ' ' 
 
 «os/l-|-cos«..4-f cos~,2;«_l] ^"^ " '^'"^''^ ^""'^- 
 
 tion, ^ . ^ . 
 
 .<4w*. tan mA. 
 
 .>.) D 1 cot \ A — tan i /I 
 
 ... ^-^^-^Uj^^I^ to a single function. 
 
 Am. cos y|. 
 
 to a fin-'le func- 
 
 23, Keduce 
 
 lion. 
 
 tan |45°4.A^l-f cot 145" + {, A 
 
 Ann. coH /I. 
 
 -'4. Hcluee 2co,,45°-f A .^, cos a6' — A ^) to a singl.' function. 
 
 2o. deduce ^"i^!.+ di - ^'"^<'" "A'. 
 
 .4?«i). cos ,.4. 
 
 to a single funetiou. 
 A/IS. sill ,4. 
 
 i ii 
 
 »4 
 

 « . 
 
 2G. i;..li;n' ^f-iir :-l">° 4 ^ ^1 — 1 to II sir<;lf nmcticii. 
 
 An/), siu i-l. 
 
 27. Keducu 1 — 2 Aii' (46" — ^ ^i to u sinj,'lc lunction. 
 
 Ann. sin .4 
 
 IV I. 1 tiin ,45° +A.4) — tnni4r.° — ^^1 . 
 
 _'fe. Jieduce -to- -",-;; /_, T -, to sin''le function 
 
 tan A5'' + I A) + 11^145-" —-h A) ° 
 
 Ans. sin A. 
 
 ("HAriER V. 
 
 IWKSK ANOI LAR I'LNCTIONS. 
 
 An inverse function is one in wliicli the indepMulent variahlc 
 ' f>e.e patjv 22i is regarded as a function of tlie variable which would 
 he dependent in the coi responding direct function; so that if, taking 
 the general functional symbol fvx\ wo have 
 
 U =fw 
 where // is a function o'i j-, the corresponding inverse function is 
 one in which .c is taken as a function of //. This is expressed 
 
 •^--=f '■.'/' 
 and is read ./■ = a (juantity whuse function is //. 
 
 Honco inverse angular functions are thosi; in which the sine, co 
 sine, etc,... are taken as tlic independent variable, the angles be- 
 ing considered as their functions. They are written accoiding to the 
 general notation id)o»e given. Thus, if « denote the sine of an nn- 
 gh', sin a expresses the angle of which a i the sine; that is 
 if sin A = ((, we liave A = sin 
 " cos A = II, " A — cos 
 " tan A = <(, " A = tan 
 And so on. They are read 
 
 -I = the a"gle whose sine is a: 
 
 
 1 422 1 
 
 .1 = 
 .1 = 
 
 And .<io 01 
 
 Hence we 
 
 an impurit 
 
 n and A, is i 
 
 uotiition. 
 
 Any relatio 
 
 ill the preced 
 
 uftlie eiinrie 
 
 'J 
 
 uiience by (^ 
 
 liecome.s 
 
 In a siniila 
 tain easily 
 
 Making cos = 
 uhtain 
 
;cii. 
 uA. 
 
 1. 
 
 1 A- 
 
 ugle function 
 a A. 
 
 dent variable 
 which would 
 that if, taking 
 
 50 function is 
 xpressed 
 
 ;li the sine, co. 
 the angles Ix- 
 •coiding to the 
 ■iine of an nn- 
 lino; that is 
 
 (422) 
 
 — 9;i— 
 
 .4 = the iinglc wlioso cosine is a; 
 A = the angle wliose tangent is u 
 And so on. 
 
 Hence we see that an angle A, which, in a =. sin ^ ia evidently 
 nn nnpurif function of n, on account of the mutual dependence of 
 
 «andA, ;sexi.n..sMdasant.y;/,<vYfunctionofa by the inverse 
 iiotiition. 
 
 Any relation which has been established -among angular functions 
 in the preceding chaptera, may be conveniently expressed by means 
 ot the w( ('('/••-•(,' notiVtion. 
 
 Thuj by (Gi we have 
 
 cos /I = ^/[i _siti^^] 
 wlience by ,422, A ^ cos ' v [1 - sin'"' A] 
 
 making a = ,sin A we have A = sin -„, aud it 
 
 sin ■'a=cos V[l-'«-^J ,423, 
 
 m a snnilMr way, making sin - a in ,23,, ,33,, ,38), (41) we oh- 
 tain easily \ » u 
 
 sin '(( ^ cosec 
 
 1 
 
 — <.ii.- I 
 
 = tan 
 
 VV~a'] 
 
 ^ cot- ' ^d'-::^a 
 
 a 
 ^ , 1 
 
 v[i-«T 
 
 (424, 
 
 (42oi 
 
 (42Ci 
 
 (427) 
 
 Mak 
 obta 
 
 ^S ^os = a in (1), ,20,, ,34., ,37,, ,44>, ,352), ,367), (351), we 
 
 cos-'„ = sin ' vXl — rr'] 
 
 (428) 
 
 li 
 
 
 [ 
 
 'm 
 
 
^ m 
 
 —94— 
 
 I . 
 
 = sec 
 
 -..-.V[iz:f:i 
 
 = tan 
 
 _ nnf.~' — 
 
 cot 
 
 = cosec 
 
 1 
 
 
 = :jcob 
 
 2 tan' 
 
 ■V(^) 
 
 (429) 
 (4301 
 
 (431) 
 H32) 
 (43S) 
 
 (434) 
 (4251 
 
 Making tan = </ in (16t, (27), i32^ (376), (377^ ^185l, we obtain 
 
 (436i 
 
 tan "'a = cot ' - 
 
 = cos 
 
 gtan^'a = sin 
 
 1 -f ((' 
 
 (437) 
 
 (438' 
 
 (4391 
 
(429) 
 (430) 
 
 (431) 
 ('132) 
 
 (433) 
 
 (434) 
 (485) 
 
 Ji, we obtain 
 
 (436) 
 
 (437) 
 (438 
 (439. 
 
 --U5- 
 
 — liin ' 
 
 1— a'l 
 
 \\+a') 
 Makiii;,' cot = a m il2., 21), i30,. wh ol,t,.in 
 
 1 
 
 cot ' a = ti 
 
 tn 
 
 sin 
 
 V[l + a1 
 
 = cos ,— -- — - - 
 
 v[i +«•■■] 
 
 Mnl<inj,'R(.c = « in ,10,, {\A\ 
 
 soc 'd = cos 
 
 = tan - V[^/-'— 1] 
 
 .1 
 
 Milking cos,. c -^ u in 5 , il8i 
 
 co.s(>c '(/ — sin '' 
 
 It 
 
 = cut 'V["'-'--l] 
 I'y (159l we hive 
 
 1 — tan A tan i? 
 
 Making tan .1 - ui and tan ]i = n. wu obtain 
 
 i44fi| 
 
 (441) 
 
 (442) 
 
 (443) 
 
 (444) 
 
 (44S) 
 (440 
 
 ■•447) 
 ;448) 
 
 ■\ 
 
 i 11 1 
 
 I' 
 
 i 
 
^ 
 
 fill) 'hi -t- tun '/( •• . till 
 
 t , 
 
 1 — /;i/( 
 
 .Siiiiiliiily liy (ICOi: 
 
 I'.v IGli 
 
 fi*-<lC2> 
 
 Hy (IMi 
 
 tiin ' m — tan ' « = tail 
 
 uot ' ;;( -|- (!ot ' /t — eoi ' 
 
 :^taii ' (( — tan 
 
 iii—n 
 1 + mn 
 
 lilH — 1 
 in 4" M 
 
 , mn -f 1 
 n — >(i 
 
 ■2n 
 
 I _ <r 
 
 l!y ,lUGi 3sin '(/=;sin i.-'.a — 4a'i 
 
 And so on lov any otln r I'dalioii. 
 
 :4HI 
 
 itftdi 
 
 tni 
 
 (15: 
 
 (3r»;( 
 
 ('UAl'TKIi VI. 
 
 liKi.ATioNs iiKi'wi;i:x A.NorLAU FuNciiovs and tiik 
 SinKS OF A Triano'.k. 
 
 In l.his Chapter wo shall invfiti^'ati,' thi' relations wliiuli hold Im 
 twetm the sides of n Triangle and the i'iinctionn ol' ilH Aiigli's. Tlif 
 I'ormtilio I'xprt^sHing these relations are. the basis ol' Practical Trigono. 
 motry, an will be seen in ("hapter VfFT, wIkmc they will be applied 
 to the sohition of Triangles. 
 
 Lot the Angles of a Triangle bo dein^ted by A, li, C, and the side 
 respectivtdy opposite to those ujiglesby a, h,e. In a right-angled Tri- 
 uilgle, A will always denote the rigiit angle; and therefore </ will 
 
 stand for I 
 ing the rij 
 Accord i 
 domonstra 
 <Ieri.e ar.n 
 the sides a 
 
 The radi 
 
 right-angle 
 oppo.iito to 
 At U as coi 
 a radius /y( 
 bitrary len 
 cribe the n 
 Let fall on 
 perpend icu 
 Then PL is 
 of the arc 
 (the radius 
 en as unit 
 angle B mc 
 By the 
 
 or 
 
 which is t 
 wo have 
 
 From (464 
 
 ,('■ 
 
,44y 
 
 III — n 
 1 +mn 
 
 lU)Oi 
 
 II n — 1 
 
 
 III 4- » 
 
 
 nn 4- I 
 n — >/i 
 
 :tr.i 
 
 ■2a 
 
 t4r)'j 
 
 a — 4(ri 
 
 [:):<'> 
 
 )NS AMI rilK 
 
 tioiis which liold hn 
 IS of ilH AiigU's. Thi^ 
 ! o1' ri'iiclieiil Trigoiiu. 
 . tlicy will bo iipplii'il 
 
 A, B, C, and tho sid.' 
 Inai'ight-unghidTvi- 
 lUid tlu'iot'orc '/ will 
 
 —97— 
 
 4and i'ov the hypotenuse; h and c will reprpsnnt tlio sidos, incliid- 
 in;; the liglit angle. 
 
 According to the method followed in ihis book, wo shall first 
 demonstratu a fundamental proposition from which we shall next 
 dnri. analytically all the relations that may be conceivetl between 
 the sides and the angles of any triangle. 
 
 Fundamental Proportion. 
 
 The radius, or unity, is to tlio sine of either acute angle of a 
 right-angled triangle, in the ratio of the hypotenuse to the sii' •• 
 oppo.^ito to tliat angle. 
 At JJ as centre, with 
 a I'adiuH liO of an ar- 
 bitrary length, des- 
 cribe the arc OP. 
 Let fall on A lithe 
 p(^rpendicular PL. 
 Then PL is the anie 
 of the arc OP, or, 
 (the radius being tak- 
 en as unity,! of thn 
 angle 5 measured }»y ih" arc OP. 
 
 By the simil ir tii;;n,';VfM PRL and Aiy\ we hivp 
 
 rr-.PL:: liO-.Ar 
 or 1 .• sin/? : : a '. h ;454) 
 
 which is the fundamentftl jnoportion above enunciated. Similarly 
 we have 1 : sin 6' : : a : c 
 
 Rinh* angled Trinngle. 
 
 From (464i we deduce; 
 
 sin B = — , or sin fj = — (455) 
 
 a a 
 
 'n ! 
 
 i 
 
 ''M (I 
 
 \J 
 
 U 
 
/^. 
 
 [IK- 
 
 wluiW liy "Sl>, -iiici' // .mil f tw 'vjiujilciiu iii;irv: 
 
 I'JS // = , Ut ■ IJ-i ' ' — 
 
 (1 It 
 
 rlOGl 
 
 I ■ I 
 
 Diviiliiij,' ylbi \>y »•">•» : 
 
 /' c 
 
 UU Jl := ■', or tan (,':= , 
 
 i\r,l 
 
 Divitliug \456) by i455j: 
 
 I < 
 
 cot Ji = -r> or cot C = - 
 h c 
 
 Taking tho reciprocal of (456, 
 
 u a 
 
 soc i> = , or see C = -- 
 c b 
 
 Taking th3 rociproc.il of (455': 
 
 cosec D = — , orcosuc C= -- 
 i; 
 
 (45*1, 
 
 |409i 
 
 i4GJi 
 
 that ia; in a rujM-anyleU tri iijle the Hiiie o/euch acute umjle is <:■ 
 qaal to the oppunitu side divided by the h'/jiottttune. 
 
 'J'he Cosine of tack acute anyle in vi/ual to the adjacent aide di- 
 vided by the hi/putenuse. 
 
 The i angent of each acute amjle ts equal to the opposite iide 
 dicided bi the adjai'eni "m:. 
 
 The CUamjenl of each mute angle in eijaal to the adjacent aide 
 diuided by the ojiponite one. 
 
 The Secant of each acute anjle is ejua/ to the hypotenuse divid- 
 ed by the adjacevt side. 
 
 The Cosecant of each acute angle it equal to the hypotenuse di- 
 vided by the opposite tide. 
 
 Converting tiie above rosiilts into profiortion givis, the riiuius be- 
 ing unity, 
 
 1 : mx li \ : II : h, or 1 : sin C : : a : c 
 
 1 : CQn B : : '( : c, or 1 : cos C : : a : b 
 
 1 : tun B :■.'■: b, or I : Uu : : b : c 
 
'( acute umjle is c- 
 
 se. 
 
 adjacent aide di- 
 
 tke opposite iidv 
 
 the adjacent sidi: 
 
 /I'/potenuse divid- 
 
 he hypotenuse di- 
 
 vs, the riiuius be- 
 
 — »0— 
 
 1 : cot B : : b : c, 
 1 : Bee fi : : e : a, 
 1 : coseo B : : b : a, 
 
 or 1 : cot C : : c : 6 
 or 1 : sea C : : b : a 
 or 1 : C0800 C : i e i a 
 
 From the same results we derive successively 
 
 h = 'X sin B 
 — a cos U 
 = cian B 
 = c cot 
 _ a 
 sec C 
 
 a 
 
 or c = a sin C 
 
 z= a cos B 
 
 = bUnC 
 
 = bcotB 
 
 a 
 
 cosec B 
 
 aeoB 
 
 a 
 cosoc C 
 
 (46J) 
 (462) 
 (463) 
 (464) 
 
 (466) 
 (466) 
 
 that is, in a right-angled triangle, either side equala: 
 
 1. the product of the hypotenuse into the sine of the opposite angle; 
 
 2. the product of the hypotenuse into the cosine of the adjarent acu. 
 te angle; 
 
 3. the product of the tangent of the opposite angle into the other side't 
 
 4. the firuduct of the cotangent of the adjacent acute angle into the 
 other side; 
 
 5. tl^. hypotenuse (Unided by the secant of the adjacent acute angle; 
 G. the hypotenune dioiaed by the cosecant of the opposite angle] 
 
 Frgin i455i, (466), 459), (460) we derive also: 
 
 
 sin B sin C 
 
 c 
 cos B 
 
 cos C 
 
 // SCO C = (• HOC B 
 
 b cosec B = c cosoc C 
 
 (467) 
 
 (468) 
 
 (469) 
 (4701 
 
 %- 
 
 1 *» 
 
 I . 
 
 I 
 
 '"< 
 
«♦»•' 
 
 — Imi. 
 
 t . 
 
 1, eiffiff ni'l' i/ii'iiliil III/ 1 1 I Kitir if II. r ilhi/li Of'/ioniti' to that 'iJi . 
 •J. eillnr xiili' iliiiiilitl I'll tlif mshif (if tlif mute unylenljarfiU to 
 
 tlint Miilf, 
 ',\. iitliff niitv multiplied Inj tin' ivntnt at' tlw acute unfile ailjacftit 
 
 til till it niile; 
 4. eitlur sidr inuUiplii'd hij the cosecant of the umjlc oppotite to 
 
 that xiitr. 
 
 Suljstitutiinj i.'JSTi in i387i givuM 
 
 2 Hiu' i /y = 1 
 
 (■ a — (• 
 
 wlieiic 
 
 Siiiiiliivly 
 
 nil! i li 
 
 Hin i (J 
 
 -':\^ 
 
 
 j 
 
 i471. 
 
 iiUu 
 
 Similarly 
 
 /( — ,■ - 2a sin' \ b I 
 
 I — /- = -in sill' h V 
 
 i 
 
 i47J) 
 
 Substituting (466i in (371 1 gives 
 
 whence 
 
 tan i /y ="-=--^ ] 
 
 tan i C 
 
 ,-«-<> i 
 
 c ) 
 
 i473» 
 
 Substituting (456) in llS'J) gives 
 
fiinln: 
 
 ^itl^ to that «/'/'; 
 
 !(//»; itljiii'intt to 
 
 (tnf/le ailjivriit 
 tjlf opposite til 
 
 (471 
 
 i472) 
 
 (473) 
 
 ~U)\ 
 
 co« 2/i = 
 
 <-*~h* 
 
 whcnen, Hinco 2B = 90*— (C— /?|, 
 
 sm iC — i?) = 
 
 a* 
 
 and by il79l 
 
 2hr 
 cos iC — li = • ,- 
 
 and bv l475i and i476i 
 
 tail, C— //I •--- 
 
 U'-f 61 l(! — h\ 
 
 (4741 
 (478) 
 
 (47CI 
 
 Ohllipii' lUlijIiil Ti lilii'ilf. 
 
 In tho triangli! AliC, lit fall tho poriuiihliciilar t7'on Ali, then 
 thu two right angled triauglo8 AVt and Jit'J' Qiva by (40 1) 
 
 Henco 
 
 Whence 
 
 Similarly 
 
 Ci> = 6 sin A 
 
 CP = ^' gin B 
 u ain B =: b sin A 
 a ih = sin i4 ; sin B^ 
 a : c — ainA : Bin C ^ 
 b : c = sin 5 : sin Cj 
 
 (477, 
 
 ri 
 
 M 
 
 r 
 
 I 
 

 _l(i2— 
 
 t . 
 
 that i>»; in <>ny trinnfilr, the sidps are proportional to the sirie^ of 
 the opponitc aiKjIes. 
 
 Thcso thvce prorovtions being written in one, give 
 a : h •.<- = sin A : sin // : sin C 
 
 (4781 
 
 which Kives 
 
 sin A _ sin A 
 
 h = a. 
 
 in B sin B 
 
 sin 
 
 sin A sin C 
 
 sin C 
 sin B 
 
 sin 6' , sm C 
 r = ti. -. = 
 
 J 
 
 (4791 
 
 l!y Con 
 
 Coiuparin{{ 
 
 and conversely ' 
 
 a . a 
 .sin A =9in B - = sin (7 -- 
 
 sin B = sin yl- = sin C - v 
 
 c c 
 
 sin 6' = sin i4 - = sin 5 ~- 
 
 a b 
 
 i478 may, also, be written thus: 
 
 It. 
 sin A 
 
 ain/y 
 
 sin (7 
 
 (4801 
 
 (481) 
 
 Tliore relations hold true when the perpendicular CP falls with- 
 out tVe triangle as in ABC, because by Lho supplemental relutiuii 
 b(!t.woi'n the angle ABC mv\ the angle PBC^ we have 
 ..in A IJC - .in PBC. 
 
 S'lnilnrly 
 
 <^nveitint 
 
 It 
 
 II 
 c - 
 
 t'lat is; in u 
 ''«'-■(', in thi 
 imite to t/ii 
 
 whence the 
 
to the sinen of 
 
 ve 
 
 14781 
 
 (479* 
 
 (4801 
 
 (4811 
 
 c 
 
 licukrCP falls wHli- 
 iupiilemeiital relutiui) 
 wo liiivc 
 
 -li/J . 
 
 r.y Comiiosilion and division, tho first equation cf (477: yivt-s 
 a -\- li ; fi — h — sill .1 -f «iu Ji sia A — nin Ji 
 
 n -J- h sin ^4 + sin Z 
 a — /' HJu A — .sin B 
 
 Comparing with (339t: 
 
 a + b _ ton ^{A + Ii)^ 
 
 Similarly 
 
 a — b tan J iA — Ii\ 
 
 a + c _ tanijj_ -fgi^ 
 a — <: tan i ( ^ _ 
 
 r4-h _ tan ^ it' -j-fi\ 
 
 *!—b ~ tilll i ( J~/T 
 
 (48J) 
 
 <'onverting into proportion ;,'iv 
 
 es 
 
 '/ f /. : .,■ - h r. tan J. A -f A'l fin I^A — H) 
 'I I /• : f,' — ,. - tiin A 1.4 ~ C'l tan I tA — O] 
 c + /; : <; _ ft r^ tan i (/1 4-^1 tan i{A—B\ 
 
 438 
 
 that 
 
 mat 18; in a,n, friun;//,; the mm of any two skhg i, to their differ- 
 ma; in th- ratio of thr tawjent of h,df tlw suw of th,' amjlea op- 
 imite to thexe svlcs to f/w tangent ,.f half their d iff. rmoe. 
 
 whence thci pmctical formula': 
 
 IrP \ 
 
 %i 
 
 4* 
 
 I 
 
101 — 
 
 ,, — /, .- .( -\- h .- 
 
 t;ui 1 .1 - n ^ 
 
 ll 
 
 II — (' = ((« -f ''■• 
 
 tilU js A -; li 
 
 tan f 1-4 4- i^l 
 tau-i ^G -^i)^ 
 
 anil convi'ioely 
 
 Uiu J 1-4 — 11 - tan I A-\ U 
 
 l,\ 
 
 <i-\-h 
 
 tan i (^ — C'! r:r tan h A -\ 11. ^ , - - J- 
 
 (( -f- *■ 
 
 tan J, .(J -11)=: tjin A Cf /i . '"-^- -'• I 
 
 It is evident tlrvt in ilu; i;ist two iiKurcr wo liavo 
 
 ASh 
 
 .4.sr» 
 
 AB= AP+ PB -. b cos A -\- a cos B 
 or i45 = AP— PB - b C03 A — a cos (180° — B) 
 
 = ll cos A ■\- II cos /y, by (86i 
 
 Hence the relation is true whether the perpendicular falls with- 
 in or without the triangle; we have finally 
 
 a ■= b aos C -\- i\cos B 
 b = p cos A -\-a cos C 
 c = a cos B < b cos A 
 
 i 
 
 i48Ci 
 
 that is; in fnii/frloni/le, curb xiile In e/juul iu the mm of thiprmhiil 
 if I'urii if ilir ollicrf' into llie cusiiic if ihe itnijli' irhir/i if mulics 
 
AAi\ 
 
 ,4sr)^ 
 
 0°- B) 
 
 by (861 
 
 licular falls with- 
 
 i486i 
 
 mm ofiJiiprodurl 
 'e which it wulun 
 
 — KjC— 
 
 irifli tfip/irxt xide. 
 
 From tho third equation of i486) we obtain 
 
 u ooa B = c — b cos A 
 whence n' cos' B = e'-^- b' cos' A — 2 he jos A 
 
 and since by (481 1 «• sin' B = b* sin'-' A , we obtain by addition 
 
 a^coa" B + sin^ B) = ,■' + b' (cos' /1-f «in' ^ i— 2br cos ^ 
 
 whence by [«] y,/^/c 13; 
 
 Similarly 
 
 whence 
 
 a' = c' -{. b* — 'Ibc Qoa A \ 
 li^ = a' 4- ,;•■« _ 2ac cos B I 
 
 b>-\-c'- a^ 
 
 |487) 
 
 C08^=: 
 
 cosZ? = 
 
 •Ibr. 
 
 'lac 
 
 cos C = 
 
 a''4-i'_«2 
 
 (4i'8i 
 
 2nb 
 
 which express eAe C-nme of each angle in terms of the sides. 
 
 Adding both members of the first equation of (488) to, and sub- 
 tracting them frjni unity: 
 
 I + cos ^ = 1 -f- 
 
 6' + c»— «» 
 
 'Ibc 
 
 _ 2fe + y -|-o'. 
 
 Jb + cf—a' 
 
 m 
 
 k^ 
 
 m i 
 
 I 
 
 '^ 
 
Hi 
 
 — ion— 
 
 and 
 
 1 — cos /I — 1 — 
 
 [II -\- 1) -f-c) \li -}- (' — ll\ 
 '"'Ihi- 
 
 26c 
 
 26r 
 
 a' — [I I — 1-\* 
 
 2hc 
 
 - [a — 1?^ — «)] [«4-l^--^l3 
 ~ ibe 
 
 _[a — h-\-c] \a-^b — c\ 
 
 ["] 
 
 ["] 
 
 But by[a] (pa<7^13),8in' yl = 1 — cos .4 = (l+coaA)(l — coaA 
 
 Thoroforo sin A = 
 
 4oV 
 
 making .j -}- /; -f c = •!.•<, it beci>nu's 
 sin' il = 
 
 , 4s (s — a) (,s- — h\ is — c) 
 
 ,,. ., , . „ n 4s (» — rtl (s — h\[s — i;\ 
 Similarly am- 3= 1- - i 
 
 1" f .' — 
 
 ain' C = 
 
 ab' 
 
 t489i 
 
 Hence 
 
 Similiiry 
 
 whence 
 whence 
 
["] 
 
 ) — cS\ 
 
 c\ 
 
 [^•] 
 
 8 AHl — COS 4: 
 - 61 (rt -f /> — '•! 
 
 (481) I 
 
 -107- 
 
 t'hencB 
 
 sin A 
 
 be 
 
 V[»{s — n){s-b){s — c)]) 
 
 '^'^ ^ ^ "^'^1"***"'" (*— ^) (*-- <^'] 
 
 ^ 
 
 8in 
 
 ^ ~ ■^'*/'^* '*~^'* '«-*>'«- c)] 
 
 (490) 
 
 which nxpress thn sine of each an'jle in terms of the aides. 
 fiy (3571, 2 sin' J /I = 1 —cos ^ 
 
 _ {a-t-c — b){a-^b — c) 
 •2bc ' 
 
 making a-^ b-{- n = 2s 
 
 by [y] 
 
 = ' ^^'-b)2 ^s — c)_ 2is-.b)(s^.) 
 2bc 
 
 b,: 
 
 Hence sin' ^A 
 
 tiA - I" — *) (s — c) 
 
 6c 
 
 1 
 
 Similury sin* ^B = -i*-ZL«!illz!i 
 
 «c 
 
 whence sin* ^C r= ( ^ = « > («— 6) 
 
 ab 
 
 («1) 
 
 whence sin Jyl = // -^ - 6Ug - d V 
 sin ii? = //!iZI^.Il^) 
 
 (493) 
 
 
 M 
 
 • ^ 
 
 
/y^ 
 
 I . 
 
 whicli LfjinsK till' .o'lf "/ lidlf of ciic/i mujlr in fcrm-i of thr sidcH. 
 2 cos'' ^A = \-\- cos A 
 
 (/(-f-t + ct (64-c — a) 
 
 ibc 
 
 -, by [«] 
 
 making a -\- b -{- c = 2s 
 
 be 
 
 Hence cos* hA — 
 
 Similarly cos' ^B = 
 
 coa'iC = 
 
 ■lo is— a) _ 2.S [h — (.i) 
 
 8 {8 — a) 
 be 
 
 s \s — b] 
 uc 
 
 s (» — c) 
 (lb 
 
 Whence cos 
 
 cos 
 
 cos 
 
 
 (493i 
 
 (4941 
 
 wiiioh exprcfis the cosine of half of each anrfle in tn-ms of the sides. 
 Dividing (492i Ly (41)41 gives 
 
 which ,'.,-} 
 ■•<'(/es. 
 
 Wvidin 
 
 which exp) 
 Allien. 
 
 The 
 
 reci 
 
of th I' sidi'x. 
 
 [a] 
 
 (493) 
 
 (4941 
 
 lormsof the aides. 
 
 tun }iA = 
 
 —109— 
 
 l,<) h)(!i — (\ \ 
 
 V\ s{a—a) ) 
 V \ s (« — h) f 
 
 (49.') I 
 
 (496) 
 
 which ,v/»;vy.s the. tanrnf «f half of ea,-h artgle in terms of the 
 Hides. 
 
 Dividing (494i by 1 492, gives 
 
 cotA»= /L^/'J''-^^ 
 V\us — « (,s — «, / 
 
 V\ [a — a, [>i~h\) ^ 
 ^vhich e.pres. th. cotangent of h^af of each angle in tern, ofth 
 
 The reciprocals of (494) and (492) give 
 Vv is — , of 
 
 (497) 
 
 SIC |C : 
 
 V(x,.?— ,.y) 
 
 J 
 
 \ 
 
 
 1 
 
 1 
 
StSSimSSfSHSi^i 
 
 Mi 
 
 -110 
 
 Vfv 
 
 :^i 
 
 ."H . 
 
 '1 
 
 ;(! 
 
 Ml 
 
 which vxpri'-^s tho sflr^mt i>/li(t'/ii'cii,-/i niX'jh in 'it ,is <,f f!,, .<;,ii\s 
 
 An.l 
 
 COSOC 
 
 COdOC 
 
 V \ I" — /', l.s —c, f 
 
 ifl = // ii'^ \ 
 
 V \ «— al \s — CI / 
 
 \/\[S — a} {« — /,) ) 
 
 l49», 
 
 cosec ^C = 
 
 which express theconitmnt «f hnlf ,if mcj, anjh In Ivrmsofthcmdi^. 
 From (481) we have 
 
 a-\-b _«[n A-\- sin li 
 e sin C 
 
 sin ^0 COM tjU 
 
 observing the supplemental relation of 
 (A + JJ) and C; and, therefore, the comple- 
 inimtal relation of ^{A + U) and ^C. 
 
 _ cos ^ C cos i(i4 — li] 
 cos f 1^" -f li) cos ^ 
 
 q4-^^ _ cos ^{A — yii ■ 
 c COS ^1^4 -fVj'' 
 
 <' + c _cos^u4 — r'l 
 
 /' ~cos ^iA +~a) i" (499i 
 
 7^ + c _ cos J,/? — Ci 
 
 that is 
 iSimliarly 
 
 Addiuo 
 first oijuut 
 
 H 
 
 01' __Jl 
 
 and — - 
 I* 
 
 u-4- 
 or ■ — ' 
 
 also 1 -f 
 
 
-Ill- 
 
 ii„ 
 
 in '''>• ,is of ijic Aiilfi 
 
 in ffli'maoft/ie silica. 
 
 lomontal relation of 
 horefore, the comple- 
 -f- JJ) and ^C. 
 
 By ibllowing a snn.bv procoMs, wo obti 
 
 « — /^ _ ain l[A — li) 
 c ~ sinXAH^ 
 
 ^ — c _ sin ^lA — C) 
 
 illU 
 
 h~, 
 
 sin 4 yl 4- //) 
 
 _8in |(^-~C) 
 sin il^+'c'i" 
 
 (500) 
 
 Adding unity to, oi; subtract 
 
 lirst oi 
 
 •ng it from both inonilwrs of the 
 
 luation of ^499, and the Hrst of ,600), wo havo 
 
 1 + 
 
 a-{-/j 
 
 aoH^tA~B) 
 
 I + -:^- 
 
 COM ^\A -f Bi 
 
 or 
 
 and 
 
 or 
 
 also 
 
 or 
 
 +_'' + '■ _ 1°^ hA -hB)-^ cos A(/l — 
 
 (■ 
 
 
 
 
 COS )^[A 
 
 t; 
 
 — 1 
 
 = 
 
 cos 
 
 COS 
 
 \\A — ii\ .:. 
 
 
 — c 
 
 = 
 
 cos 
 
 l\A~IJj — 
 cos i{A 
 
 1+"- 
 
 -h 
 
 ^ 
 
 1-t 
 
 sin|U_j! 
 
 /?! 
 
 + JJl 
 
 COS ^(^ -f- 2?) 
 
 ain ^(^ -f ^, 
 'i±J' -/* = i!^i:i4 + //l-f sin ^' 
 
 '■' sill ^(ii-f iy.i 
 
 /t — /yi 
 
 4» 
 
 '^ 
 
 and 
 
 r-— 6 
 
 1 = !i5.ii^ ■- -fi) 
 sin |u -jTsr 
 
 - 1 
 
 '■ sin i(.l + ;»! 
 
 ^ 
 
"Wry t-i'-lii-maSIESBCS! 
 
 «'*^^ 
 
 -lU' - 
 
 • , 
 
 uiakiug I = i(« 4"'' "f* '') '" '^"' "'"* mt'iiilins of tlif iivcnciliiif; i! 
 i|iiations; and tiiinflfonnin;^' tlin luiiiu'i'iitoiH ol' lln' wccuud nu'iiihurs 
 liy t2fi4l or |r>59i, and t\w dononiinntorH by (81) uud |8Ji, woobluin 
 
 n cos ^A lios A/y 
 
 c ~ sin y ' 
 
 It — (-• _ siu ^A sin ^JJ 
 
 I' ~ siu ^(.' 
 
 ti — b _ sin ii4 cos ^JJ 
 (? COH \< ' 
 
 *M — u COS ^A siu ^Ji 
 
 i; ~ cos |C' 
 
 iflOli 
 ,502) 
 (503) 
 (5041 
 
 Performing tiie the same operation on the otlier e(juiitions of(499i 
 and (450), wo obtain 
 
 s _ cos ^A cos ^( ' 
 
 h 
 
 
 sin 
 
 hJi 
 
 s — h 
 
 — 
 
 sin ^A 
 
 siu ^C 
 
 II 
 
 sin 
 
 iJJ 
 
 s — a 
 
 = 
 
 sin ^A 
 
 cos iC 
 
 b 
 
 cos 
 
 iJi 
 
 S—C 
 
 = 
 
 008^4 
 
 sin iC 
 
 b 
 
 cos 
 
 i^ 
 
 s 
 
 = 
 
 cos ^B 
 
 cos ^C 
 
 II 
 
 sin 
 
 iA 
 
 H — a 
 
 
 sin -J/i 
 
 sin ^C 
 
 (5051 
 
 (506) 
 
 (507) 
 
 (608 1 
 
 i5oyi 
 
 i510| 
 
f \ 
 
 p lin-noflinR "' 
 cuuil iiu'iiiltur.s 
 (8il, woobtaiu 
 
 1 501 1 
 
 ,50'.') 
 
 (503) 
 
 — li:}- 
 
 .s — h _ Miu \H con \(J 
 o COS ^A 
 
 s — (■ _ COS f,n sin JC 
 u CUM hA 
 
 CIIAl'TKR VII. 
 
 *' 
 
 ir.ni 
 
 (512i 
 
 t 
 
 5041 
 
 jviationsofi499i 
 
 (5051 
 
 ,6061 
 (SOT) 
 
 (508 1 
 
 (5oyi 
 
 ,510) 
 
 Tbiuonometiik! Tables. 
 
 liofore pi'ociioding to ai)]ily thn foriiiiiliL' doduced in tlui lust clmp- 
 ter to tiic nuuKU'ical conipi.tatiou of tri;inglnx, wo shall uxpliiin how 
 to tlnd tho niimnrii-til vulue of Angular Functions. In fact, thn vari- 
 ous Trigouoniftric (,,)iiantitieH, thi; sine, tlic coaino, &o., ar« hut ab- 
 stract nuniher.s r'-proscnting tlm coinparativc Icn^^tli of certain lines. 
 Wc have already siscn, in Chapti-r 111, how to obtain the niinieric- 
 ul value of Angular Functions in paiticular casc.-t i^/c /;'(;/»'.s 14 , 
 45, and 46. i Wc shall sluu-, in this chiiptcr, how the iiunilicrs re- 
 presenting the angles of ail li ■.ijics oi' magnitude, from 1' up to 90°, 
 can be ascertained. 
 
 When the radius i laiien im tiie unit of measure, it is manifest 
 thattho trigonoiuetrie (.UiUililies are simply the nuuihers exjiressing 
 the ratios of tho angular functions to unity, [nvt pwjc. 12.1 And as 
 the sine and tlui cosine of any arc never exceed tho radius, but, 
 on the contrary, sin 90° and cos 0° excepted, are all less than tho 
 radius, it follows that the Sine and ( 'owiiU! of all arcs or angles in- 
 chuhid between .° and 90° are leas than unity, and, therefore, are 
 expressed by decimal fractions. 
 
 Tangents and Cotangents which vary from to infinity, may 
 have any value greiitor or less than unity. 
 
 k 
 
 '■*\ 
 
^% 
 
 — IM — 
 
 t , 
 
 Hocauts fllHl ('o«0<'nIlt^ lllivr tlli'if \alllf \\>\ ;ill.\ illxli' "t^i lillilr 
 
 iinigiiilmli', ^{rcnlci ihiUi unil\. 
 
 Till' iiuiii'hm'm coni'H|ii)iiiliii;; lu tli".siiic, comiii', »Vl'., ci nil iiii;ilis 
 IVoiii r to W, wlifti iirrim{{i'il .nui-oiilinj^ to tln' iwci'inling onlf. ul 
 til"' iiii^lrH, I'oriii the Trii/iiiunirtrii' Tnhli's. 
 
 'I'lii'ii' nil' two iiriiini|ml ti'i>;oni>iiii'lii<' tililrn. 'I'lii- lirst contniii- 
 tlii' iiiiiiicriciil viilui's of Hiuc, iVn., iih ti.'I'oii' stiili'il, uiiit in imiHi'iI 
 tin* 'lohlv iif Nnfiinil Sims, lyr. Tln' .tcroiul coiitiiiiiH t'li' I,o;,m 
 iitlini« of the DUinbursi in thr tiist tiiMi'. niul is ohIIimI tin- Tahl,i,( 
 LiDjitrifhiiiir Siiif", ij-c. 
 
 Wu HiippoHo hnre thiU tlin Mtiiil«nt ImiiIiimuIv liCiiuaiiitiMl witlillu- 
 ntituri! and tlio uso of thw coiiiinon fiiMi' ol lonaiitliniM of iiuiiiImms. 
 
 Tlio tlvRt opi'i'iition to be |n'ifoiiiii(l in coiii)iutiiij^ a tiiMn of na 
 tuval eiui'H anil cosini'4 of all till- aii|;li!H ditl'i'viii^' li.v I' iniii(iiiiil 
 rant, in to ascertain the valni' of tin* sini' of tin- lowi-st iiiij,'li', ov 1'. 
 liy niHan« of this valuo and nouw fonimlif Ix-foin cHtalilishml, lln' 
 valutui of siueH and cosint's of all tlm otlmr anf,'lt''<, ain oasily ealcii 
 laU^d. 
 
 Tht' radius being unity, I In- si'iiiioirciiiufi'roiice is kaown wc 
 gcoiiM'tiV' to he tH|uul to 
 
 '' = .■» .1 H.'i'.t^tir);;... 
 Theiidort', dividing " by tln' iiuuibi-r of niiniitcs containiMl in tlir 
 ieniiciroiind'erencc, gives thi' valu*' of one niinuto of arc, tliat in; 
 
 180 X 60 
 which is the hmgth of one uiinutti of arc, and i« denoted by K fof 
 brevity. 
 
 Wo shall now prove that no ditfurencn lu'twoeu the sinoof onr 
 minute of arc and the arc itself, exists within the ninth place of de- 
 cimals, in the alwve result. 
 
 It is manifest, by the insi)ei;tion of tlie tigure, page 12, that i4 A' 
 is greater than /IP, and that ^/' is greater than JWI; that is, llie 
 sine ol an arc not exceeding !)()°, thi^ arc itself, and its tangent 
 are in ascending order of magnitude. 
 
 The limiting value of the ratio of an arc to its sine, wh(*n the 
 arc is indefinitely diminished, is unity. For, in the first phice, we 
 
 whence 
 
-n:.-. 
 
 i\ iiii^'li' III' II tiiiitii 
 
 , vVc, rinli Mii;^ltH 
 wc'i'Udiiit? onlc. Ill 
 
 'I'llC lilMt fOlltllill'* 
 
 ilcil, aiiiJ if< iMilli'il 
 
 (■DIltlUllM t'll' I,o;,M 
 
 callc<l till- Talili'xf 
 
 iciiiiiiiiitfd with till' 
 ritlmiH of nuiiibi'iK. 
 iitiiig a I a' lit* of nil 
 f by r ma utiinl 
 lowest anj,'li', ov 1'. 
 
 W CHtilhlirtllHll, till' 
 
 fs, iiin (>a.sily ciilcii- 
 »)iiee i.i kaown wc 
 
 itis coiitainctl intli'' 
 itu of luc, thatiw; 
 
 0807... = A' 
 
 ia lU'Uoted by K for 
 
 ,vt)nu tlici sine of oni' 
 lie ninth [tliiei) of df- 
 
 I'fi, page 12, that it A' 
 m I'M, that in, the 
 self, and itts tangent 
 
 its nine, when tlu' 
 in the first place, we 
 
 have 
 
 III the Hejoud jdujo 
 
 ''in A A 
 
 — - — li'\i //tun - — 
 A A 
 
 ■in A Hill A 
 
 — ~ — ijrrii/rr tllilii 
 
 A 1,111 A 
 
 since tan A is },'ivator tlian A, an Henn befon. 
 
 „Hin A 
 
 rheiolbre tlie value of — - Ileci.ssaiily lien belweeli 
 
 A sin A 
 
 -—-and - — - or coh A. .v*/- i8,. 
 A tan A 
 
 lint the liiuitiu;,' value of the firHt ratio, is evidently I; and tliat 
 of the second, also, i.s 1, siuce cos U = 1. Therefore, the limiting 
 
 , -siui4 
 value of -^ iijuHt, also, be unity, that is; when A is made less 
 
 than ;>ny a.si;,Miable i|iiantity, ure.|iial to 0, we have 
 
 sin A _ 
 A 
 
 whenct! 
 
 •sin A 
 
 ur the sine of an are having no assignable value, is efjual to the arc 
 itself. 
 
 By assigning any finite value to A, aniall though it bo, the rr tio 
 becomes less than 1, since sin A is loss than A. And the greater is 
 the value assigned to A, tlie greater will lie the difference between 
 sin A and A; and the lesser the valutf a.ssigned to A, the lesser the 
 tliiference betweeq sin A and A. 
 
 i 
 
 *«< 
 
J ita Mg 
 
 -1 IC— 
 
 It iii.iy ivjW K- sliov.ii (li;i! till' sine oi'iui arc is ;,'icati'i' tli^iii llic 
 .IVL- iiiiiiiuisli"(l ]>y L'lii' t'uiirlli of tin' (Uilii' oi' lli:il arc. 
 
 F'.'i !lu' laai;\):! oT all air. li 'iuL;' gri'aU r tlini tli" iivc. we liavc 
 
 litii .'..l -^ ■■ ijrrdfor tliii'f \A 
 (•us .',,4 • 
 
 wllCll'.'i' 
 
 ill ;',.! iii-ratir thini }.A cos A.4 
 
 J5ut liy i^.^T^. sin A = 2 sin !.l cos Ll 
 
 SiikstiiiUiny; hA cos LI for sin hA in tliis ('(niatiuii, it liccoinos 
 
 .sill A ijrftitcr fli,ni 2L-1 cos' ■! A 
 
 or sill ,1 ijriiit.T ill. Ill A cos" lA 
 
 or sill A i,riiilrr iliiiii A A — >iu-' },A^ 
 
 ])iit siii''c ;iii ar" is ..iPHtcr lli in its .sine, wc li iv.', hy .'iubstitutiu"' 
 l-L'lr i'ur sill' \A ill tli;' last ini'iiuality, a fortmri 
 
 .sill A rr.'in'.r tlnni A[\ —lA ''] or A —\A wliicli 
 was to lie ipi'ovcil. 
 
 Now niakiiij,' A -- 1', v.c liavc con.sc(}ii<'ntly, 
 
 s,;i 1 ,,'n;ifrr l,',uii 1' — ^ IV' 
 
 Ami tiikiiiH- the iii.^t six dcciiiials of A', tli" iiuiiiciiciil value of 
 one luiniUu, increasing ilic sixtli decimal liy unity, we have k fur- 
 tiuri sin r greater than [0.()()>)-_>1)08SSJ08 ... — ^^ .OOOJS)!,']; 
 that is, iierlbiiiiiiii; the arilliiin'tii^Ml opetatioiis: 
 
 sin ]' orratri- t/i ri .00()i>!)i)S87922r)... = //, 
 
 'J'hus we have iouiid two decimal fractions, A' and Jf, between 
 M'liich i.lie V ilu ■ ol' siu r must lie, since .sin 1' is less tlian the lirst 
 
 ( 'onstaiitl' 
 in i'>\:',t 
 .■ill r- 
 
 .■•;ii i2' -I 
 
 sill .••)'-|- 
 siu I -f- 
 aiid so on. 
 COS I r -j.- 
 cos i2'4- 
 oos ( ;r-f 
 
 cos (.f-j- 
 
 aiid so 01). 
 
vc. 
 
 ■ !ivc. wi' have 
 
 11, ithoL'oiiiPs 
 
 I', liy !<ii1)stituting 
 1' A —^A wliicli 
 
 miut'iicil vahu' ol' 
 .y, we liave n fur- 
 f.OOO.'Ol.'j; 
 
 1225... =-- //. 
 
 \ and JI, butwetiu 
 ■< less than the first 
 
 -117- 
 
 (A'lan.Um^atc.r than the second (//i; and tlieso two fractions a'^reo 
 -" tlK'.r lirst .ight figures, therefore, we n.ay say without error, Uiat 
 
 sin 1'= O0()29(t8S... 
 
 since the .lillWence between tiie two members of this equation be- 
 gins only beyond the decimals assumeil. 
 
 Jf a great,.rnpi„.oximation be required, a larger number of deci- 
 mals could be assumed in tlie approximate value of ^, or the value 
 ot one second oould be fouud lirst, from which the value of one 
 minute; may be easily deduced. 
 
 The value oicos T, i.s found by i6 , that is; 
 cos r - ^/[i_,sin I'l 
 
 = VI. J 
 
 0.00029088] 
 
 - 99!)'J9!)!)r)77... 
 
 The Sine and Cosin.' of 1' being known, we may obtain the Sine 
 aiKlLosineofall angles in (he quadrant, by meanJ of the formuke 
 r.o-^) and ,2o4), which, after tue transj.osiliou of th.^ second term 
 become 
 
 ■ sin M + /y , = 2 sin /I cos H - sin ; .1 - /,•) ,513/ 
 
 cos \A -f li, -.-.: -2 cos A cjs n - cos '.A _ B, ,514, 
 
 Constantly making /y = T'lnd , I =.- 1 'V 'V r i.,> • , 
 
 . , , , ■= ' ' '.-,.>, J , iVc, successively 
 
 III i.il.-.i a,:d .■>! I , w(! obtiiin 
 
 ■■'■' _'^'+ I'^ -■' -"' 2' z^ 2 Cos I'sai r— sin 0= .000r)8I776... 
 .-^.11 \-l -I- r, - .sill 3' ~ 2 cos r sin 2' - sin 1'= .0008726G 
 sni ;:/-!- 1 , _- ,si„ 4' ^, eos 1' sin 3' —sin 2'= .001163r)i32" 
 mi 1 -f- r, = «in ,y _. -2 cos I' sin t' — sin 3'= .001454440..." 
 and so on. 
 
 eos,l'-|. ri = cos 2'= 2 cos 1' cos 1' — cos = .999999830... 
 
 cos ,2'4- I'l =: cos 3' = 2 CO-, r eos 2' — cos 1'= .999999619.".". 
 
 cos I 3'+ I'l = cos 4' — 2 cos 1' cos 3' — cos 2'= .999999323... 
 
 cos i4' + I'l = co-i .V = 2 cos r 008 I ■ -cos 3'= .999998942..'. 
 and so on. 
 
 1 
 
M. 
 
 .118- 
 
 I'.y till' same procnsscontiimcd, we slnlloliUiu the vulurs of llii' 
 .-siiK^s mill cosines of all iiughs differing l)y 1' from (P to OO''. 
 
 The tangents, cotangents, secants ami cosecants may next be cal- 
 culated H'spectivcily, 1)y the formiilie ill', d") , r2"i, (23). 
 
 It must he remarked that an error committed in calculating the 
 sine or cosine of an inferior angle will entail errors in the same 
 functions of succeeding angles. Therefore, the comjuitist must have 
 means of iletecting and correcting errois. 
 
 The mo.'5t simple means consists in calculating the sines and co- 
 .sines of certain angles, by an independent process based on some 
 established property, and comparing the results thus obtained, with 
 those obtained by the general method. Thus, for exami)Ies, we have 
 seen how we can find the values of the functions of 45°, 30°, and GU°. 
 
 It is not necessary to continue the jjrocess beyond 30°, because 
 by the formula' (283) and (286i, we have 
 
 sin (30° + 5| = cos i^ — sin |3<H — Bi 
 cos (30° -\-B\ = cos (30° — Ji) — sin B 
 
 where we see that the table may be continued beyond 30° by the 
 simple subtraction of the sine or cosine previously found. Thus; 
 
 sin 30°1' = cos r — sin 29^ 59' 
 sin 40° = cos 10° — sin 20^ 
 and so on. 
 
 CO. 30°r = COS 29^59' —sin 1' 
 
 COS 40° = COS 20° — sin 10° 
 and SO on 
 
 Moreover, it is not necessary to continue this last process beyond 
 45°, since the sines and cosines of the angles above 45° will be res- 
 pectively the cosines and sines of their complements below 45°, 
 
 Let it be stated that m consequence of the tedious operations 
 
 which the ab'. ve process re<]uire8, other methods are resorted to in 
 
 the actual computation of tables, which recjuires Calculus. In a 
 
 part later on, we shall deduce liy means of Calcidus infinite series very 
 
 rapidly converging, from which the UMoh are calcnlated with great 
 
 1. For 
 into 
 
 2. For 
 with I 
 
 3. Wht% 
 into ( 
 
 io»;y| 
 
 Tile rodl 
 there is ail 
 uf 10; ant 
 4"al (loficj 
 
 We }ia\| 
 t'iking thf 
 
II!"- 
 
 rl 
 
 he val\ics of till' 
 (P to W. 
 may next LimmiI 
 
 u caUniliiting tin' 
 ■ova in the suiiw 
 iputist must liiivc 
 
 the rtiues and cu- 
 ss baswl on HOiiif 
 iu» obtained, with 
 ■xamples, we have 
 45°,30°,andGO°. 
 ond 30°, becaiisr 
 
 sin ii 
 
 ,.you(l 30° by the 
 found. Thus; 
 
 last piocoas beyond 
 ove 45° will be ivs- 
 iients below 45°, 
 tedious operations 
 Is are resorted to in 
 ires Calculns- 1" ■' 
 IS intinitt^ aoriea very 
 jalculiited with j^veiii 
 
 iiciiuiK^y and nmnli less lalior. 
 
 'i'lie tallies of I.ogaritliiiiic Sini's iVc., ('(intuius tlie lij{,'aiillniis of 
 tlie numbers in tlie tal)les uf ii.ilunil sines &c' Jlut as most of 
 these nunibci's are less than unity, tiieir logarithms would consc- 
 «(nentiy have negative indices. This inconvenience is avoidfcil by 
 multiplying the Trigononuitric Functions calculated in the above 
 manner, by a very lar^'e nund)er ^vliich undeis them all gi eater than 
 unity, so that their logaritlims be positive numbers. It is evident 
 that the same operation being perlbrmtd upon all, their relative 
 value is not altered, 
 
 The number thus employed is by common consent settled at tlie 
 \alue of 10'" = 10 000 000 000; and the results give the values of 
 the trigonometric functions as if caculated to a radius being 
 10 000 000 000, which are simply the logarithms of each incre.xsed 
 by 10, 
 
 Thus the natural sine of 3 ' is .052336, wliosi' logarithm is 
 _2.718oOO 
 to which add 10 
 
 8 .718800 
 which is the logarithmic sine of 30° in the Uddes. 
 
 Of coui-se, in calculations, we must evidently allow for that in- 
 crease, and we have tlie following rules for practice; 
 
 1. For each loi/uril/mir/uitvtion introduced with thcj'oaitiri' alijn 
 into an expression, nubtract 10. 
 
 2. For each liKjari/hjun; /mution introduced into an expyesnion 
 with the nayutivt: aii/n, uUti 10. 
 
 3. Whtn themmi' numlier of lo:jarithtnic/tinction)< vW introduced 
 into an ejcprmsion a ith hi,th the posit ine and thi: neyatioe siyns, 
 10 in completely nei/lected. 
 
 The reason of the above rules in evident, since, in the fii-st case 
 tliere is an excess of 10; and in the second case, there is a deficit 
 of 10; and in the third cast!, an cfpuil excess is balanced by an e- 
 qual deficit. 
 
 We hav(! stated that the logarithmic tables may be calculated by 
 taking the logaritlims of the uumlurs in the natural tables, and in- 
 
il 
 
 crcisiiin till' ill, lex liy In. IJut it will lie sc, mi licn'iil'tci- tli.il tlicy can 
 
 111' UlllculiUcil illi/i'iii'ii<(i'llt/i/, tll't is, willluul till' Usi' ut' till' fcllilcs o' 
 
 tilt! ^satiiral Functions, hut iiy iulliiiti' wiii's. 
 
 A-i to tlio Uvi- of the 'I'lijjoiioiiii'liii; Tiilili'S, \vi' icIVi' the sluili'iu 
 to l)')oks oont-iiinin^' thi'iii, when' all fXiiliiiMtiuiiK rcLitiii;,' to tln' 
 iiiiiimi'V of iisiiij;' tlirm, nrc fiiUv ■4i\' ii 
 
 CHAPTKIl VIII. 
 
 Solution op Trunoi.ks. 
 
 lu every Triangle thovo are six I'lcimaits. imuioly, the throe sides 
 find the tlivoe angles. 
 
 The solution <if <i trlamile consists in calciiliitin},', liy means of 
 the foi'imilie given in CliivpthV VI, the unknown oleinents, from tlie 
 relation which they bear to tiiose wliicli are known. When three 
 olemcnts are known, one of theui, at least, being a side, the others 
 can always bo determined. 
 
 By (46 
 
 whenci 
 
 By (46 
 
 whenct 
 
 By Gee 
 Example. 
 
 Rii/Jit-(tn;/'i il Tn'fiHi//''. 
 
 In Ejght-angled Triangles, the riglit angle biiug always known, 
 any two other parts one of which i.; a side Ining given, we can de. 
 tormino the rest. 
 
 All cases that may occur in ;)iMclice, are reduced to the four fol- 
 lowing: 
 
 1. When till' liypoli-niis ■ :,iiil oui' of the anitu angles are givi'U 
 
 2. When one side and en • of tlii' ;;ciiie angleti an' given. 
 
 3. Wlirii the liy[)otenuse and ou" side arc givm. 
 
 4. Wlu.'n the two sides are givi'ii. 
 
—121— 
 
 Case 1. 
 
 Given the hypotenuse a and the angle B. 
 
 (ilv, the throe sides 
 
 ici'il to till' fourlbl- 
 
 By (461), 6=asini5 
 
 whence log 6 = log a + log ain B—IO (515) 
 
 By (462), c = acoBB 
 
 whence log c i:: log a + cos 5— 10. (516) 
 
 By Geometry, (7=90° — 5 (517) 
 
 Example. Given a = 276, and B = 57° 23', to^find h, c, C. 
 By 515-, (516), and (517i, we have 
 
 + log 275 2.439333 
 
 + log sin 57° 23' 9.925465 
 
 — 10 1 0. 
 
 = log ^ 2 .364798, giving b =-- 231. 63 
 
 + log 275 2 .439333 
 
 + log cos 57° 23' 9 .731602 
 
 — 10 10^ 
 
 = log c 2 ,170935, giving c = 148. 23 
 
 ('- 90° --57^ 23' = 32° 37' 
 
 i 
 
 Vi 
 
Case 2 
 
 Given oiii- sidt; li, and one nciitc luij^le IJ. 
 
 J{y (404) r - h cot li 
 
 wlitiuoe loi,' c = log /' -f log cot /^ - 10 lolR) 
 
 JiV (tO"t « = -; rr 
 
 whence log a - log /^ — log sin 7? -f- 10 (519) 
 
 By Geometry, C = 9 ° — Ji (r.-JOi 
 
 Example. (liven h = 125, and li = 38° 41', to find c, n, C. 
 By |618', ,51'J , and (520), we have 
 
 -flog 125 2 .096910 
 
 -I- log cot 38° 41' 10 .090544 
 
 — 10 10^ 
 
 = log c. 2 .193434, giving c = 156 . 15 
 
 + log 125 2.096910 
 
 — log sin 38' 41 9 .79.5871 
 
 -t-!0 10. 
 
 — log a 2 .301019, giving 199 .50 
 
 C' = 90°— 38° 41' = 51° 19' 
 
 S .B. Kb is ({iven with tlio ailjiiccnt iiiiglo C, then we opomto by tho formnln) 
 (lll)aiil(l'H) 
 
 Case 3. 
 
 Given the the hypotenuse a and the side 1i. 
 
 K.\aini)l( 
 
 l!v (!( 
 
1 i ,!=J 
 
 ing c = 156 . If) 
 
 — 1-J3— 
 
 ]5y (455) and (4801, sin 7^ = cos C' = 
 wh^mcc lo;;- sin B = log cus (' = log /> — loga-f 1^ '■'''■^l) 
 
 r.y Geoiuotry, c = V[«'— ''*] = -v/[(« + ^l !« — ^'t] 
 whence log c = ■J[log la + h\ -f- log \n — 6)] (522) 
 
 Exaiiii)!.'. (Jiven a ~ 053!) .76, and b = 2!)36 .91, to find B, C,<: 
 
 By (521 1 and i.')22i we have 
 
 4- log 2i);iG .01 3 .407890 
 
 — log 0539 .70 3 .815502 
 
 4- 10 10. 
 
 = log sin /? = log ens C 9 .0,52328, giving 
 
 i? = 26M1' G",aud C= 03° 19' M" 
 
 And aince a -{-h— 9470 .07, and a — b= 3002 .85 
 
 + log 9470 .07 3 .970650 
 
 + log 30O2 .85 3- .556646 
 
 2)7 .533302 
 
 = lot,"' 2 .76605], giving 
 
 .^ = 6843 .49 
 
 ۥ 
 
 
 
 i^ 
 
 '! 
 
 i 
 
 .^. 
 
 orntc by tho formuln) 
 
 Case 4. 
 
 Given the two sides /; ,inil 
 
 By (457| ami (i58i, tan B = col C= - 
 
 whence log tan B = log cot O = log h — log c -f- 10 (523) 
 l!y Geonuary. </ = ^/[/<^ -f. c^J ^^2^] 
 
»A 
 
 — iL't- 
 
 ) .. 
 
 but tlif last foniiulii is iijt siiilahli' l'uilu,i,'iiiitliiiiic coiiipiitation, 
 
 K.\IUlll)li'. (■iv'Cll /' r: 70.'!, f - l.':>. t'J lill.'. li, C, il. 
 
 I'.y TiL';! ami '.")•_' I, Wf liiivK 
 
 -f log 76.1 -J f<s36(;i 
 
 — log 123 2 .(IfSll'.Mir* 
 
 -f ID 10 . 
 
 - log tan // = log cot 6' ^ \') .I'JMV) ^vhi^ H - n'.51">7" 
 and r^ fl' 8' 3" 
 
 <^= v['7fi-'5'-' 1 •.'?.] 
 = v[5«J-->+ 1''>1-'!)J 
 
 - .^ [fiona.vjj -771 .s:^ 
 
 The str.deut iniy see i'rom llie aliovi! imiilts, that in any on-; of 
 the fo I reason" of tiight-aii'.;l(Ml Triangii;.*, each of the three unknown 
 elements uanbe determined by a pioiier ehoije of the fonnula from 
 its direct relation to tin given eliiiK nis. 'Jhereloie, when only 
 one part is rc(Hiired to be known, it cin bi^ in every in.stance, de- 
 termined ind(4)end''ntly of the rem lining unknown parts. 
 
 'Since the wines of angles whieh are nearly •.♦0-' and, also, the co- 
 sines of very small angles, dilfer very little from each other, as may 
 be seen by the inspection of the tables, it follows that a small angh? 
 cannot be found with great acciii'acy from its cosine, and should, 
 thvU'efore, be determined from its sine; and conversely, an angle 
 wliieh is nearly 90° should be comjiuted from ils cosine In 
 either case, the angle is always computed with great accuracy from 
 its tangent or cotangent. Thus when the hypotenuse a and one of the 
 sides (■ arc nearly equal, in which case the angle JJ is very small 
 iiud llie angle Cnearly 90°, the forniuhn-lTl or (473i should be used 
 inite:ul of l.')6i for computing li. 
 
 The formula '472' maybe usi'd with advantage to compute a — r 
 when B is very small, or a — '» when C is very 'niiall. 
 
 By the formula 47.51 or 176', ('■- IJ may he fouml v;ith great 
 
 ", Ij, 
 
 % 
 
 whi 
 also 
 
 ;viu. 
 
 -V. R. h 
 'hiru iii.g (I 
 
<4. 
 
 b 
 
 nic computation. 
 
 - i,y ^\"u' 
 
 tluil in ;iny on-; of 
 till! lliiu*! unknown 
 li th'> i'onnulu Ivoni 
 VL'loie, when only 
 (•very instivnce, di- 
 own [iiuts. 
 )' ami, ivlso, Ih.' co- 
 1 each other, us may 
 iTs that a small angli- 
 cosine, auil should, 
 onversely, an angle 
 loni its cosine In 
 great accuracy fi'oni 
 luseuamloneofthc 
 gle n is very small 
 °473i should he UM'<1 
 
 ft,> to eomiiuti' (I — '• 
 
 ■iuiall. 
 W' found with gveat 
 
 — ''2r.— 
 
 accumcy wlicn c and h are nearly equal. 
 
 Obliijue Angled Triangles. 
 
 All ca8(>s that may occur in practice, are reduced to the four fol" 
 lowing. 
 
 1. When two angles and one side are given. 
 
 2. When two sides and the angle opposite to one of them are given 
 
 3. When two sides and their included angle are given. 
 
 4. When the three sides are given. 
 In the tiamgle 
 
 AnC\ let the an- 
 gles be denoted by 
 A, li, V, and the 
 sides respectively 
 opposite to them 
 by the correspond- 
 ing small letters - - 
 
 A -^ 
 
 _______ ._j \B 
 
 (I, b, r. 
 
 Case 1. 
 
 Given two angles, A, />', and one side //. 
 P.y Geometry; C = 180° — i / -\- li\ . 
 
 By i47yi 
 
 wlienco 
 also, 
 
 ■.vhence 
 
 , sin A 
 It - h- — — 
 sm li 
 
 log a = log h -f log .sin A — log sin B 
 
 , sin C 
 Sin £ 
 lof, c = log i -f log sin C — log sin B 
 
 (62C) 
 
 (526) 
 
 (527) 
 
 N'. B. If the pivon sMe he li.cliicleil hotwoen the two given angles , then the 
 IMnx 'ii.ge neeesKar'.iy requires to be determined, first, tt.s In (526). 
 
 
 i 
 
 'M 
 
.|:.'r.- 
 
 r,\:l,l|.ll tiivcll .1 - !'•'■ 
 
 It - •;;; i."*'. '- -•■ ■-'''"'• '"j i'""' 
 
 c, .i, 
 
 r,v i">-'.")i. i"'-20i .111. I .•'•J7i w' liiiVk' 
 
 J.. 
 
 k---'... 
 
 -Ho^;siu-ir 2.. 
 
 r Z.T I so' - !'.)■ ■-'."> 
 
 
 -f .1 
 
 5' lS':r:::GC' W 
 
 lu- sill (i:5' !.>< 
 
 =- Ic 
 
 , <.l .ll.'fJillS 
 L' .:'>(Ui!>lin,j^iviii;,' <( 
 
 = -iiVJ .77 
 
 + lo- 1275 
 
 .4:\!)3:$;i 
 
 + lo-siii(iO^ 17' 9.iMi;«:ii> 
 
 1:2 JO-Jiif)."* 
 - lo- sn. 03' 4S' J.uriiiillS 
 
 , I I'.ithi. ^iiviii;. 
 
 Cii.^i. -2. 
 
 = 2f<l .1)7 
 
 ( Invn t wo si.lrs. ,v ;mil h:m\ tin' :ni -1" .1 opi-o-^itc !'• oii,. of iImmh. 
 
 r.v 4 SI I 
 
 ■<iii r. .-■ SI 
 
 n.! 
 
 I'lu'iirc li.in-sin /.' — lo.L' sill 
 
 ^■1 ..'-l,,- /, — lo-t '(-- 111 
 
 ■ i'.s. 
 
 Jt mm 
 
 in the pi 
 
 »'unt trial 
 
 «'<!, oach 
 
 tho cond 
 
 lom; and 
 
 i.f anibig 
 
 of this 01 
 
 • lid'oiont; 
 
 and til! HI 
 
 1'oth trinn 
 
 "ladc cqii 
 
 I'h'niontar 
 
 f'orp, tho s 
 
 nil cnsns ^ 
 
 iin^le shoii 
 
 It is ovi 
 
 1. whon t) 
 is greati! 
 gle mu8 
 given 
 
 2. when 
 glos mx. 
 
 MOM!OV( 
 
 •Icitonnino 
 Kxnniplo, 
 
 V.y (51 
 
 4-iog 
 
 + log 
 
 l',\ ( II (IllH'tl'.V 
 
 Ct-. ISO - .4 4- /; 
 
 Till' ini;j,lc ^'lifiuL.' (Icti'miiiicil. lln' tliinl siilc '• is caily Ibuin 
 
 iv<> in 
 
 (.',1V- 1 
 
 — loo 
 
 = loff 
 
/ 1 
 
 I'it. lo I'll"* 
 
 ; = CO" A"' 
 
 .t I 
 
 t<l .tJ7 
 
 to.m.'ofUu'Ui. 
 
 „ -10 ' 
 
 52><' 
 
 ,r)2!t' 
 
 ,. is r;vily i'oUll'I 
 
 — 1-.'7— 
 
 It nmst h« remarkod that 
 in tlio prosout cnso, two dillu- 
 rcnt triftnglcM may be form- 
 ed, each of wliich will satisfy 
 the conditions of the prob- 
 lem; and then the solution 
 is ambiguous. Tlio roason 
 of this ambiguity is that two 
 ililfiMcsnt and unciiuiil triangles may lie constructed, having two sides 
 and tin angle oppj-iito to one of these sides respectively equal in 
 both triangles, as seen in the annexed tiguro,whore ISC' and li'C are 
 made equal to eiicli other. And since the angles D andZi, are sup- 
 plementary, and the sines of two supplementary arcs are equal; there 
 fore, the sine found for li, may bo the sine of either the acute an- 
 gle A no, or of its snpiilement, the obtuse anghi AB'C. Hence, in 
 all cases where it is not possible to a,sceitain whether the require<l 
 angle should bo ncuti' or ohtutic the solution is ambiguous. 
 
 It is evident that there is no ambiguity: 
 
 1 . when the given angle is acute, and the side opposite this angle, 
 is greater than the other given side, to which the roquiivd an- 
 gle must bo opposite For this last angle will be loss than the 
 given one, and, therefore, afottiori will be acutn; 
 
 2. when the given angle is obtuse, because both the remaining an- 
 gles must bo acute. 
 
 Moreover, in practice, there is generally some circumstance to 
 determine whether the re(iuirod angle is acute or obtuse. 
 
 Examplo. Gi\ en a = 14/5, 6 = 173 .3, .4 = 41° 10' to find ^ G 
 P.y (528i we have 
 
 -flog sin 41° 10' 9.818392 
 
 -I- log 173 .3 2^251151 
 
 12 .069543 
 
 — log 145 2.161368 
 
 = log sin B 9.908175, giving ^ = 54" 2' 22" 
 
 or U = 125° :->T 38" 
 
 ,ri 
 
 J 
 
ff 
 
 4 
 
 If wi! take tlid liHi't value 
 
 C = 84° 47' 38", ami the trianglo k AUG. 
 
 It' wi- take the lucoud vitluo 
 
 V =■ 12° 82' 2i", und iho tiianglo in AB'C 
 
 lu huth uuHUH, tho third nidc /■ iaditteriuiiiod •» in caHo 1, by 
 (627 1. 
 
 CiiHe 3. 
 
 (iivon two 8idi*8, <t and h, and thuir included angle U. 
 
 liy i485, tan i 1^4 — /?) = tan J ii4 -f /i -^^ 
 
 whence log tan J (yl — //) = log tan ^ (/I -f B) 
 
 -f log (rt — t) — log ^a-^ b (830j 
 
 Tho B'^m ii4 f B} is found by subtracting the given angle C from 
 180°. i\na mo ditfjrenco (^4 — Bj being determined by tho abovu 
 formula, we have, by an algebraic relation: 
 
 IA + B, + ^{A — B) = A 
 l[A-\-B) — ^\A--B\= B 
 
 And when A and B are thus determined, the third side is found 
 flu in the first case, by (527>, 
 
 Example. Given a = 535, //— 420, C= 53' 8', to find A, B, ,■ 
 we have ^(A -f B) = ^(180°— 53° 8'l = 03' 20 
 also, (!-{■!) = 535 -j- 420 = 056 
 iind a — h = 535 - 420 = 115 
 
 (;i 
 
 'I'll,. 
 
 liH' Jliuh 
 
 'liu form 
 •I'Mij ain 
 
 or 
 
 And 
 
■VJU - 
 
 and by (530i, + log luu C2'' 2fl' 10 ,30090!) 
 
 -flog Il» 2 .OGor)i)8 
 
 12 .:Jtil(i((7 
 — log OOri 2 .1180003 
 
 = loKii/l--/y 'J.38!f!!i|,"-'ivin;{ 
 
 \:\° .32' 2rt" 
 
 wli-nuc // = 03' -.O' 4- 1.3' 32' -.'5" = 70° ns'i/I" 
 
 and C = 03' 2C' - 13" 32' 25" = 49° 53' 35" 
 
 To find the Hide c proceed m in the example of ciwe 1. Wo may 
 
 however determine c by the fomnila ,499) from which wo deduce 
 
 <'vjden*!y 
 
 »' 
 
 
 i 
 
 '^<l 
 
 Cnsc 1. 
 
 bird Bide is found 
 
 (iiv<,n tho throe .sidos, a, b, c, to liiid tlm nnglos A, Ji, C. 
 Tliu iinj,deciiii l»c found by the funuuhi' 190), (492i or (i94 ; but 
 the most convenient method of oomi.uling them is derived from 
 tliufunnui.e (i9o., tho first of which, whon both num3riitor and 
 d'Uij uinitor are multiplied by (,y — ai, bocjmas 
 
 V\ »(s — ui' / 
 
 or 
 
 tan f,A = — i- J('^^!^~Ailz:i£)\ 
 
 And denoting tlie second factor l)y /■, wo ha. 
 
 t.tn ^A - 
 
 S — (■ 
 
-];j(i— 
 
 l!^^ 
 
 tun hJi= -~- 
 
 .-i — h 
 
 tiiii J6'^ — -- 
 
 *■ — r 
 
 \vJi( iiuc Jug lim lA — 1(1 + log /.■ — log (.V — u) (,")31 1 
 
 log tuu hB= 10 + log /.• — log (.V — l>\ (r)3-2 1 
 
 log tiiii -^ r • _ K ) -I- log /• _ log (jy _ ) , ,-)33 1 
 
 Exaiiiiilo. t;ivi'U a = VA), I, - 140, c = 130, to find A, B, C. 
 
 a = l,-)0 -flog 60= 1 .778151 
 
 b - 140 -j- log 70 = 1 ,845098 
 
 '-■ - i^ + '"o 80 = 1 .903090 
 
 2 ) 420 5 .5-^6339 
 
 .V = -210 — log -210 = 2 .322219 
 
 '^^'•'i^^i't:'' f — a= 60 2)3 .204120 
 
 a - /> ^ 70 = log /,• = 1 .6020GO 
 
 s~<'= 80 
 
 whcnci! by i'i31l, i532), uud (533) 
 lU-f log /, 11 .602060 
 
 — log6() 1 .778151 
 
 - lo- taiUyl 9 .823909, giving i/1 = 33° 41' 24" 
 
 10-f-log /■ 11 .602060 
 
 — log 70 1 .845098 
 
 = log tan y; 9 .756962, giving yy= 29° 44' 41 
 
 10+ log /, 11 .602060 
 
 — lcg«0 1 .903090 
 
 — log tiiii IC 9 .698970, giving 
 
 hU= 26° 33' 57 .5" 
 
■ i;,i - 
 
 Whoucc, Ity taking the doublo ol' tliesi' results 
 
 A = 67' 22' 48" 
 h = 5^ 29' 23" 
 C=53° 7' 49" 
 
 
 (53 1 1 
 (53-21 
 (r)33) 
 
 ai\A,n,C. 
 
 !151 
 i098 
 K)90 
 
 )339 
 
 >-219 
 
 11^ 
 !0G0 
 
 U = 33° 41' 24" 
 
 ;yy=29'"44'41 
 
 Vmjication A-^n-{-V=mP 
 
 Use of iSuOiiiliary Anjii:s, 
 
 When in the solution of a problem, there is met with an expies- 
 sion not suitable for logarithniie cuniputtition, such un expression 
 may be made an by introdnciuy into it an angle, which Is then called 
 a iubaididry angle. A subsidiary angle is, therofore^ an angle which, 
 although not immediately connected with a problem, is introduced 
 in it by the computist to facilitate his compulation. 
 Thus, if in Case 3, of oblique angled triangles, the side c only is re- 
 quired, it is desirable to have some Method of computing it, inde- 
 pendently of the two angles A and B. And as by (487) we have 
 
 c' = a'' + 6' — 2«6co8 C 
 
 this formula could be used advantageously for computing c, were 
 it not excluding the use of logarithms. To resolve it into factors, we 
 have by equation [a] I'tKjc 13 and by (182). 
 
 c» = (,,* 4- b^\ ico.s* IC -f- .sin' i^i — iob (cos' i^C— sin" jCi 
 = (rt -f- /»' sin'^ AC -f i« — b)'' cos'' ^C 
 
 = (a + h\' sin' iC [1 + (^jT^cot ^c)'j 
 
 Assume tan a; = . . cot \C, 
 
 i 
 
 a + 6 
 
 (534) 
 
 20° 33' 07 .;")' 
 
■■:a(f-.-;*'t:vi.-.'^!. 
 
 then c' =: (rt -\- bf sin* ^c {1 -f tan* x) 
 
 Hence <• = (a -f- 6) sin Jc sec « 
 
 or c != (u + i)— *- (535) 
 
 cos X 
 
 whence log c = log (rt + <>)-)- log gin ^C — log cos a; (536) 
 
 Example. Uiven a = 535, b - 420, C = 63° 8", to find c 
 By (534) we have 
 
 log tan a; = log « — b\-\- log cot \C — log (a -f b) 
 
 that is -flog 115 6.060698 
 
 + log cot 26" 34' 10 .300999 
 
 12 .361697 
 — log 955 2 .980003 
 
 = log tan X 9 .3t;l694, giving 
 
 X = 13° 32' 26' 
 
 2»row by (536), we have 
 
 + log 055 2.980003 
 
 + log sin 26° 34' 9 .650539 
 
 12 .630642 
 — log cos 13° 32' 26" ... 9 .987758 
 
 = log c 2 -642784, giving 
 
 c = 439 .34 
 
 Area of a Plane Triangle. 
 
 We shall close the preeent chapter by giving an expression for 
 <;oniputing the area of a plane triangle, corresponding to the four 
 
 
-1:53— 
 
 )8 X 
 
 to find c 
 
 (535) 
 (536) 
 
 -log(« + M 
 98 
 
 197 
 i03 
 
 fl694, giving 
 = 13° 32' 26' 
 
 80003 
 50639 
 
 30642 
 
 87758_ 
 
 42784, giving 
 439 .34 
 
 g an expression for 
 ponding to the four 
 
 cases of oblique angled triangls. 
 
 Casel. Having giv- 
 en one side c, and 
 two angles A, B, 
 
 Let S = area 
 
 /. = CP, tho height. Then by Geometry we have 
 
 but by (461) 
 
 Also, 
 
 Multiplying 
 
 S = icA (1) 
 
 h = b Bin A 
 h = a sin B 
 h* = ab sin A sin B (2; 
 
 a = e 
 
 b = 
 
 By (479) 
 
 Also, 
 
 Multiplying ab = c 
 
 Substituting in (2), h* = 
 
 sin A 
 
 sin C 
 
 sin B 
 sin C 
 
 sin A sin B 
 
 = c' 
 
 sin A sin B 
 
 sii'G ' Bin*(A + B) 
 
 c* sin* A sin* 5 
 
 or 
 
 A = 
 
 Substituting in (1) 8= c 
 
 BinUA + B) 
 
 c ain A sin £ 
 
 sin(il + -B) 
 
 sin A sin £ 
 
 2 sin (A -t- -6) 
 
 (587) 
 
 M 
 
 i * 
 
 ^ 
 
 i 
 
\ 
 
 — 1;{4— 
 
 whence lot? 2»S' - i' log '• + log ■•^iu A + log siu H 
 
 --hgsin yl4-i?i i5;58i 
 
 Case 2. Having given two sidca b, r, and liieii included iingle A 
 
 and by ^461) -S' - idi sin .4 *53'J) 
 
 wl euce log 2 .S' = log r + log/i -f log sin A — ]{) ^540) 
 
 Case 3. Having given two sides b, c, and the angle U opposite 
 to one of them. 
 
 In this case, it is necessary Hrd to determine the angle included 
 between h and c; and then proceed as before. 
 
 Case 4. Having given the three sides a, b, c. 
 
 By (539) S = ifh ^in A 
 
 and substituting the value !490) of sin A, we obtain 
 
 iS = ^[s [» — «) {S — 6) (* — o] (541) 
 
 whence log (S' = ^[lo- 6--|- log (*• — «) 
 
 -|-log(a — ^t + log is— el] (542i 
 
 Ex. 1. Given e = 250, A = 30° 42' i? = 43^ to find S 
 
 By (538) 2 log 250 4.795880 
 
 log sin 30° 42' 9.708032 
 
 log sin 43° 9.833783 
 
 24.337695 
 
 — log si'. 73° 42'... 9.982183 
 
 — 10 10. 
 
 = log 2»S' 4. 355512,* giving 
 
 2,S': 
 
 Ex.3. 
 
■]' lof,' sin H 
 n A + D) i5;}8) 
 
 ucluded iiuglu A 
 
 ^531)) 
 jin^l— 10 ^540) 
 
 auyle L opposite 
 
 ,e anulc included 
 
 obtain 
 
 
 }} [H — 
 
 0] 1541) 
 
 -a) 
 
 
 log iS- 
 
 f)] (5421 
 
 = 43^ 
 
 to find S 
 
 80 
 
 
 32 
 
 
 83 
 
 
 95 
 
 
 83 
 
 
 l-j; giving 
 
 — 130— 
 
 2^' = 22G73.2 
 <S'= 1133G.G 
 
 i^^- -^ A = :35, . = 1 1 7, ^ = 27°; to flind ^' 
 
 I^y(540) -flog 35 1.544068 
 
 + log 117 2.068186 
 
 + log sin 27° 9.657047 
 
 13^269301~ 
 -10 10. 
 
 = H' 2>S' 3.269301, giving 
 
 2^=1859.09 
 S= 929.54 
 
 Ex. 3. Given a = 605.3, b = 330.7, c = 402.5, to ffnd ;i'. 
 
 s = 619.25 ■"' 
 * — a =113.95 
 
 « — </ = 288.66 
 
 .V — c = 216.75, whence by |542) 
 
 + log 619.25 2.791866 
 
 + log 113.95 2.056714 
 
 + log 288.55 2.460221 
 
 + log 216.75 2.435959 
 
 2) 9^644760" 
 
 = ^"^8 4:822380rgivino 
 
 >S'= 66432.45 
 Applications to the Measurement of Distances. 
 
 By the rules given in the present chapter on the solution of tri 
 angles, we are enabled to calculate the distances of inaccessibi ot 
 je ts^ A hne connecting two points lying i„ , j.^j,,,,,, ^^f" 
 called honzontul distance, or simply, ,tii^tanrc. 
 
 IK 
 
4 
 t < i 
 
 ■■, '■■» 
 
 'i 
 'it 
 
 
 .!:',(; 
 
 A iiiir I'Oiuiriliiij; h\i) ! o Ills ill ;■ vertical ])laui' is callf'd liri'ilil. 
 
 Hut as tlit'siiiiii' i)iinciiili's rule the caloiilatioii in liotli cases, tlicic 
 l)i'iu^' no (litl'cicucc Ik l\v(\'n tlu'iii except ih it tlic. aiijrk's to bo uifiis- 
 u red live lioi'i/oiilnl in tin fii.-t iind vcilicnl in llie second, we shall 
 take tlie iiir.ttev in a ^'eueiiil re i-ecl, and loduce tlie uidinurv cases 
 that ocijur in practice to tlie tliree loilowin;^', in I'acli of which tlie 
 pvoces is cLiiductnl wholiy hy ilu- na : sdienui.t (ji' iinj^'les and 
 of one line, called lli' li.ise. 
 
 1. AVlu'n one of tlie I'oiuis io;i!i 'cleil ly a re(|n;i' d line is accessi- 
 ble. 
 
 2. Wlcn iiie two jiuiiius a: iuacce-.s.lr ■. lnit one o'.' iheiu is in line 
 
 with tha iiase. 
 
 3. "When the two I'uiul-aiv iii::Ci e-i.siMe i:i,''. ni i'.hi I in line with 
 
 the base. 
 
 Case 1. iHACho the line to be cnlcnlated, A being accossililc. If tlie 
 distance .M/> and llu' angle/) be niea.-iureii. AC is easily c<d- 
 culated by Case 1 of I'iglilanglod Triangles. Thus the heiglit 
 of a tower i.-i found by measuring a certain distance from 
 its base, and observing its angular elevation at the extremity 
 of the line measured. In the .sr.nie way the width of a river 
 iq com])ut"d by measuring a certain line on its bank perpen- 
 dicular at one extremity ton visual lim^ starting from a visible 
 object on the otliei' bank, and ob.serviug the angle B at the 
 other extremity of the base with the same object. 
 
 Ciiae 2 Let (7^ be the line to be calculated, P being inaccessible 
 
s Ciillf'd lirhlht. 
 )olh cases, tli«'i<' 
 
 IglcS to 1)0 IIU'IVS- 
 
 scc'onil, w<' >*li'll 
 . ovliiiiU'V casi's 
 , 1, of wliidi Uii" 
 ; !j1' iiii^'U'!-- iiini 
 
 ,1 lino is iici'i'ssi- 
 ly; ;hiMn is in line 
 
 IM r 
 
 in lini' witli 
 
 ;,C(M'Ssil)k. It'tlio 
 
 ,4r is easily c<il- 
 Thus the hiiight 
 lin ilistiiuce from 
 )n ivt the extremity 
 ir width of a river 
 11 its lank puri)eu- 
 iliug from a visible 
 the angle B at the 
 
 object, 
 bein'' inaccessible 
 
 —137— 
 
 but in a linei 
 with t h e. 
 ba«« -4/i. if. 
 the length of 
 ABhe meas- 
 ured togeth- 
 er with the 
 two angles k 
 
 which wd know the angle ABC^ its supplement) then the 
 angle ilC// is known by subtracting [A-\-B] from 180°, and 
 A Cm found ^rom the triangle ABC hy Case 1 of Obique 
 •angled Tfiangles; and next CP from the triangle ABP by 
 the Case 1 of Bight-angled Triangles. Thus the height of a 
 tower CP, the foot of which is inaccessible, but which lies 
 in the same horizontal plane as any two stations taken in a 
 line with it, and at any^drat^hce, is easily found by measur- 
 ing the distance between the chosen stations and the angu- 
 lar eletations of th^ tovret ttom each of them. 
 
 Ctt*iS.'£tftiBi5'bethe 
 lino'tb be cal- 
 culated, fi and 
 G being both 
 inoflcesfible & 
 -Dot IB a 1 i Q e 
 with KL. If 
 the visual 
 lines CL, and 
 BL an') the 
 angle BLC be 
 determined, it 
 
 is evident that BC may be calculated in the triangle BLChy 
 the case 3 cf Oblique angled Triangles. Measure the haaeKL, 
 theangleo CKL, BKL, CLK, and 51-^; then calculate C/v 
 
 
 ( 
 
 i 
 
'^f'ti-^'^tJt 
 
 \ 
 
 — l.ks- 
 
 iii tlu! ti'ianglo KCL, and IJL iu the tiiaiiy;k^ KBL, ]>y Ciisc 
 1 ol' 01)11(11113 anglud Triangles, and detorniinc the angle BJ^C 
 by subtracting tlie angle KLC froniHho angle KLIi; and tic 
 reii'iired elements to calculate JJC are known, TIuih the dis- 
 tance ot two objects in a horizontal plane, which are inaccts- 
 giblo is easily found. If, however, JiC be a lieiyht, a.» lur an 
 instance, a tower situated on a hill, it in then more siniiilf to 
 calculate the elevation of C above the plane KA from llic 
 triangle CKL, as in Case 2, and the elevation of the bottom 
 Ji above the same plane, I'roui the triangle BJiL, and then 
 taking the dill'erenco of the two height* thus found. 
 
 N. B. Ills iiefcssiu}- to observe tlmt In eaUiiliitliig llie li«iglil olun oliject !i. 
 bove tt liorl/.ontiil plane, w*; mii.sl add to the result found im heibie wtated, ili;ii 
 partoftlie lieiglit lyliiB Itelow the lioiizontiil nno from the eye of the obseivci 
 when nieiiMiriiiK t'>e nngiilai elevation 
 
 r 
 
 Piacticxl i'rubkms. 
 
 1. A person standing at 12 > feet from tlu base of a tower, finds 
 its angidar elevation to bo 52 ' 34'; and the horizontal line from tlic 
 eye when observing the last anj,le, intersects the tower at h^ I'eit 
 above the ground. Calculate the height of the tower. 
 
 Am. lG8.ti 
 
 2. Being on the bank of a river, I measured 463 yards by the bank 
 I found that a line at right angle at one extremity of the measured 
 distance met a line on the oppcsite side of the river, and measur- 
 ing the angle at the other extremity with the same line, 1 found 
 16° 21'. Kequired the width of the river. 
 
 Am. 13r).8 
 
 3. Two observiMs 8tnnding 8Hf> feet apart, in a horizontal 
 ground, measured at the .same time, the angular elevations of a lial- 
 loon whilst pasfsing in a line witli I)oth of them, and found .3.v 
 
 
 1;: 
 
l:i<i- 
 
 
 nine the (luglc Bl^<' 
 mglc A'L7i; imd Il>' 
 Luowu. Thus the (lis- 
 o, which arc inaccLS- 
 be n heiijM, at* lur an 
 HtheuiuorcbiuiliUlu 
 plane KA li'om lli.- 
 evationol'thebutloiu 
 ngle BKL, ami Uu'ii 
 ta thus I'ounil. 
 
 the lielglUol'uni)l>)i'<t ii- 
 iiiiul UK bc'lbre »tuluil, tliiil 
 111 Uio fye of tl>0 olJsoivci 
 
 biusu of a tower, fiinl^ 
 loiizontal line from tln' 
 Its the tower at o^ IVoi 
 |e tower. 
 
 Am. 10f<.^ 
 
 |463 yards by the bank; 
 jniity of the measured 
 Ihc river, and measuv 
 Iho same line, 1 found 
 
 and 64". Required Lho height of tho balloojx abovo tlie ground of 
 the observation. 
 
 A71S. 935. IQfcei 
 ^.Wanting to know tho perpoiidicular height of ahill,I found its 
 angular elevation, ut the bottom to bo 47'' 15', and 150 feet farther 
 on a level with the bottom of the hill, I found that its angular el- 
 evation was reduced to 30^ 4J'. What was the required height? 
 
 Aug 
 5. Being on the bank of a river, and wishing to know tho dis- 
 tance between a house and a tree on the opposite side, I took two 
 stations on tho bank at 260 feet apart; at one of them I me^ou^ca 
 the angles included between the hue connecting them and the vis- 
 ual lines to the house and tho tree, and found them to bu respect- 
 ively tiO' and '1\)"' 56'; at the other station, 1 found tlie angles with 
 the house and the tree, to be 32° and 70° 14'. What was the requir- 
 ed distance) 
 
 6. Wishing to know the height of a tower situated on a hill, I 
 measured, from a certain puiut at tho bottom of the hill, tho angu- 
 lar elevations of the top of the hill and the top of tho tower and 
 found 40° and 56°; going 200 feet farther in a direct line with 
 the tower, 1 found tho twu above angles reduced respectively to 35^ 
 Aiid 42". What whm the heighl uf the tower) 
 
 CHAPIEH IX. 
 
 Devblopement of Anoular Functio >i8 into Series. 
 
 i 
 
 Ans. 135.B 
 
 It, in a horizon till 
 lilav elovations of a bal- 
 tlicui, and found :\'y 
 
 As the investigation of Trigonometric Series.is more easily carried 
 on with the aid of Differential Calculus than by a mere algebraic 
 method, we^shall give such of its elementary principles as will be 
 
M 
 
 I'l-nuiii'd ill till' lyllowiiiH iKigcs. 
 
 O'linnil Prhiiijilf of Diffcrcnfiiitinii. 
 
 Whou i\ variable involved in any i'uuctiou is uyauined to pass to 
 another value, the amount of change, oi the (ii^envice between the 
 primitive and the new value, in called the Increment or Difference 
 of the variable, And the amount of cliango wliich the function it- 
 self is caused to undergo by the change of the variable upon which 
 it depends is called the Iw remeiit or Difference of the function- 
 Thus, if in the general functional expression 
 
 we give the increment li to the variable .i;, the function/u) receivcH 
 a corresponding increment which is denoted by D /{x), and is ev- 
 idently obtained by subtracting the primitive value of the function 
 from the value to which it is caused to pass by the increase of the 
 variable, that is 
 
 D/U-] =/(.,• + //I - /■ .n (543i 
 
 Thus, we have D sin j- = sin ix -\- h) — sin x 
 
 D cos X — cos x-\-/n — cos .«• 
 D tun X = tan '.x -\-Ji,] — tan x 
 
 and 
 
 If we assume h, the increment of the variable in any function, to 
 decreasa till it become less than any assignable quantity, or infinitely 
 diminished, it is then called the differential of the variable, and is 
 symbolized by dx, if the variable is x. And the value which the 
 increment of the function is caused to receive in this case, is called 
 the differential of the function, and is symbolized by d/(x}. 
 
 It is shown in the theory of Calculus (see my Iiitroaui,tion to 
 Differential and Integral Calculus, page 8) that the dilferential of 
 any function is always the product of the dilferential of the varia- 
 ble into a certain (pumtity dejiwiding upon the value of the fimctioii 
 
— Ul- 
 
 aud culled thp DifferrntUtl Coefficient; that is, denoting that quan- 
 tity byP, 
 
 dj [X) — Pdx, the doiTerontial 
 
 
 amed to pass to 
 ,ice between the 
 »*/ or Difference 
 the function it- 
 nble upon which 
 of the function. 
 
 tion/lJ;) receives 
 O/ix), andisev- 
 ij of the function 
 \\ increase of the 
 
 whence 
 
 df{x) 
 dx 
 
 = P, the difTerontial coefficient. 
 
 (643i 
 
 The Differential Coofflcient, for any fanction, is obtained by divid- 
 ing the increment of the function by the increment of the variable, 
 and ascertaining the value of the quotient when the increment of 
 the variable is made leas than any assignable quantity. 
 
 Multiplying the Differential Coefficient by dx gives the Differen- 
 tial of the function. But the latter may also be found directly by as- 
 certaining the value of the increment oi the function, when that of 
 the variable is made less than any assignable quantity, without mak- 
 ing the division above mentioned. 
 
 When, for an easier notation, we make 
 
 .'/ =- / 1^') 
 
 we have dy = d/ [x) , the differential 
 
 I 
 
 •M 
 
 X 
 
 (' 
 J' 
 
 n any function, to 
 ntity, or infinitely 
 he variable, and is 
 ! value which the 
 this case, is called 
 d by d/ (X). 
 tiy Jntroaui-tion to 
 t the differential of 
 ontial of the varia- 
 ilue of the function 
 
 and 
 
 dj/ _ d/{xj 
 
 ~j^ — ~~7~'-^^^^ differential coefficient 
 
 The expansion of tlie binomial term/tsc-f h\ in |543; 
 iiesames generally the following form: 
 
 /(«-f A) =f[x)-\-Ph-{-Qli'-^Iih'-ir 
 
 ■ (»«l, 
 
 where the first term isithr primitive function; and the other 
 termsin^olvetheascendingp. wersof /i,the coefficients P Q R 
 being independent of A, but depending upon / ix^. By inspecting 
 the nature of this last expression, we see evidently, fii-st, that the 
 hrat term of the expansion will disappear by the actual subtrac- 
 tion o{f[x) indicated in (643): that is 
 
" i ' ^w i^^ rf . - '•r. Ji^£tm 
 
 " 
 
 -142— 
 
 / (.-• + /.)—; u\ = rh+ Qv + ///,' + , 
 
 iini 
 
 Tlnuofoio, rtuy coustunt tiuuntitj connoctod by tho Hi;<nH -f or 
 — with ({X) will iliHappear aft^r differentiation, k'causo it in uvi- 
 lUmt that it Bhouid bo found only in the {ir»t term of tho oximnsion 
 in (»»). On tho contrary, any conHtaut factor oi t\x) would bo mul- 
 tiplier of tho whole oxinDUDiou, and .hereforo of all of it« toriim 
 and couHoquently would remain factor of the differential. 
 
 8ocond, dividing («i by h, tho exponent of h, will bo diminished 
 by unity in ouch t«rm of the expansion, and will disappear in the 
 first term. And making k leuH than any assignable quntity, or 
 equal to 0, all terms bat tho lirat will vanish, and P will be tho 
 limiting value of iho lutio, or the diHerential eoefficient. Hence if 
 /(*4-'*)be (x + A)", or any algebraic binomial having a constant 
 power, the second term of tho expansion in \m] [or the first term in 
 (n)] must be in all cases^ by the binomial theorem, ?«•»-' h) therefore 
 i» = nJ-n -'; that ii, if 
 
 y = .i'" 
 
 = /ij,« ' 
 
 If 
 
 <ix 
 
 dy — na;"-' dx 
 
 y = log ;'. 
 
 (644) 
 (645} 
 
 Dy = log ix + h) — log (.r) 
 
 Bat by Algebra 
 
 log (1 + m) = J/(m — Jm» + J«i'— ...) 
 wkere M stands for the modulus of the system of logarithms. 
 
ini 
 
 •11.1- 
 
 ho Hi^nw -f" or 
 :au8o it it* uvi- 
 tho oxpanHiun 
 would b(j imil- 
 hU of itateriUH, 
 antial. 
 
 bo dimiuiuhud 
 wippear iu tho 
 lie quntity, or 
 P will be tho 
 sient. Hence if 
 ving a constant 
 ho first term in 
 '•-' h) therefore 
 
 (644) 
 (645) 
 
 X 
 
 if logarithma. 
 
 Theroforu 
 
 /'// = 
 
 = '"(:- 
 
 A» 
 
 'ix* 3j 
 
 4- - -(- 
 1 ..'I I 
 
 
 ax' 
 
 II 
 
 t>nci! 
 
 
 and 
 
 'hj = 
 
 MUx 
 
 If, don,nin- the Naperian logarithms hy /, we h.vvo 
 we obtain, since tho moduhiM in this system is „„tiy, 
 
 '^■L - i 
 dx X 
 
 and 
 
 dij = 
 
 dx 
 
 X 
 
 (54«; 
 
 (54 
 
 »i 
 
 (648) 
 
 (549| 
 
 If// - «>^ in which « i, a constant qnantity and the exponent a' 
 a variable, we liave, Kikiug the Naperian logarithm of both members 
 
 li/ = x.la 
 
 Did'orentiutiug gives --= dx la 
 
 y 
 
 whence 
 and 
 
 dy = ydxla — la.a'.dx 
 
 ~r~ = la.a-' 
 dx 
 
 (550| 
 
 (031) 
 
 If denoting by . the buoof the Jfaporiau system of logarithms 
 we have ° . 
 
 I 
 
 i 
 
 '4 
 
~Z7j^...jsaBiBSiSsia 
 
 :#t 
 
 -144— 
 
 we obtain, since the logaiithni of th« base in any system it* unity 
 
 and 
 
 (Iff = e^ dx 
 Differentlittion of Angular Functions. 
 
 1 552 1 
 (553) 
 
 The differentiaU of Angular li'unctiions will be most easily found 
 Jf we consider, 1. that whan h is assumed to be dsr, we have 
 
 •in dx = tan 4^.1' ~ dsr 
 that is the sine and the tangent an arc infinitely diminished, are 
 actual to each other aad to are itself. For 4(t; being, when compared 
 to finite quantity, equal to 0,wj h ivtj 
 
 sin Jx 
 Undx 
 
 = cos 0=1 
 
 Wiieitce sin dx = tan dx 
 
 Therefor* {seepage 115) sin dx = tan dx ■- dx 
 
 3. that ax when connected with fitiite quttutity by the signs -f <»' 
 — mwit be rejected, aa being then considered equal to ; and there- 
 fore X muat be •ubstitut«d for x-{-dx,x-\- 2dx, x + ^x. 
 Hence, if y = sin x, we have, increasing x by h, 
 
 Dy = sin [x + h) — Bin x 
 
 = 2 cos (a; 4- \h] sin \h, by (229i 
 
 n)«king h = dx\ 
 
 or 
 and 
 
 'If 
 
 dy = '2cQ»m X ^dx 
 
 dy = cos xdx 
 
 dy 
 
 — - = cos X 
 
 dx 
 
 y = cos .1' 
 
 (»54) 
 (055) 
 
—14.1 
 
 8yHtem is unity 
 
 (652) 
 (553) 
 
 we have 
 whence 
 
 iind 
 
 If 
 
 w(- havH, 
 
 oat easily fouud 
 
 
 we have 
 
 
 iiiiiniahed, are 
 
 wher 
 
 trhen comp»fed 
 
 
 
 und 
 
 
 If 
 
 the signs -j- o>' 
 
 
 ; and there- 
 
 
 ^x. 
 
 
 \ X 
 
 
 n \h, by (229) 
 
 
 
 wlionco 
 
 («54) 
 
 
 (055) 
 
 and 
 
 Di/- cos [X -f- A) — cos a; 
 
 = —-i sin (X 4- \h) 8iu ^h, by (230) 
 
 ,lx 
 
 ~ — sin .fifx 
 
 — sin^ 
 
 (557) 
 
 // - tan .1- 
 Dii ~ tan ,.(• 4- /, I — tan a; 
 
 sin h 
 
 ~coH~u^lk,~c^-^ V (2361 
 = spc (.r + A) sec a; sin h 
 dij = sec J,- X sec x x dx 
 = sec" xdx 
 
 di/ 
 Tit 
 
 = soc- X 
 
 (558) 
 
 (559) 
 
 y = cot a; 
 ^>Z/ = cot (.r-f-/,|__cota; 
 
 — sin h 
 
 "■'=>■>' u-+/.iliir;^'*'^ '240) 
 
 — cos(.c(,r-f/,,eos,ca;sin/, 
 
 <'.'/ coscc X X cosec .r x dx 
 
 = — cosec'' av/j* 
 
 (SfiO) 
 (501) 
 
 1% 
 
 • if 
 ^11 
 
 
 *^ 
 
'■^•^-v.'T'jr "' ;iirimiiiiiiiiffir MiuniiiiwHwuiij^ijii^i 
 
 .■*/■■ 
 
 
 — 14C- 
 
 
 If 
 
 whence 
 
 // = sec ./' 
 D;/ = 803 {x-{- k) — .sec ,(■ 
 
 _ sin ! J' + i'*) sin M 
 cos IX -f- A) co» X 
 
 1)V I -'49 
 
 3 sin .<! 
 
 (1,1/ = ., - d.i: = tiiii ,(• .si'C .(' '/.fi <.")U:ii 
 
 cos- X 
 
 and 
 If 
 
 i/// sin 
 
 (ix cos''* ,/• 
 
 = tan X sec .r 
 
 .".G." 
 
 y — cosec .r 
 /)// = cosec u; + /i' — cosec 'x 
 __ — "2 cos \x + ^'(i .iin i7( , 
 sin w4" /i siii •^' 
 
 by (251 1 
 
 whence 
 
 d!/ = 
 
 — cos X dx 
 
 sin' x 
 
 — = — cot a; cosec x dx (564i 
 
 and 
 
 dy _ — cos a; 
 
 dx 
 
 am* X 
 
 — cot ./,' cosec X 
 
 = Diferentiation of Inverse Auyular Fauctions. 
 
 (565 1 
 
 The Differential Coefficients of Inverse Angular Function are most 
 easily obtained by the rule that the Differential Coefficient of amj 
 inverse function is the reciprocal of that of the eorrespotidim/ 
 direct function. This rale is proved in the Introduction to Differen- 
 tial and Integral Calculus, ipago '20|. i'hus 
 
 = / ' i.ci, an invorie function, 
 
 *h80 •«=-/^(y),thecorrespondingdirectfunc- 
 
 tion, 
 and therefore, representing by P the value of tiie ditierential coei'- 
 
-147— 
 
 - .sec ,(• 
 
 \} sin ^h 
 co» X 
 
 1)V (L'4'J 
 
 tilli.c .si'C .1' 1I.1; nUlii 
 
 sec .r 
 
 :.G3. 
 
 — coscc ic 
 
 ^^- --- 'by(25]i 
 I sin J' 
 
 ficiont of a- = iy), we have, for that of y = f-' i^) 
 
 ^vhea 
 
 ce 
 
 Hence, if 
 we have 
 
 dy 
 dx 
 
 _ 1 
 P 
 
 
 dy 
 
 _ dx 
 
 
 y 
 
 = sin- 
 
 X 
 
 X z 
 
 = sin y 
 
 
 dx 
 
 ^=co8 //= V[l-sin»2^] =[l_^j 
 
 whence 
 
 and 
 
 dy 
 
 dx ^[i_a^j 
 
 dx 
 
 (666) 
 
 xjwnvvi m/ vc»* \17»J^P 
 
 
 
 ■ V|l-^^] 
 
 
 (567 
 
 sec .f (565 1 
 '''unctions. 
 
 If 
 
 we have 
 
 dx 
 dy 
 
 // = cos -'a; 
 - sin yz=~ ^[1 _ cogj^j _ 
 
 -[1 
 
 -^'] 
 
 Y Function are niobt 
 
 Coejficient of any 
 
 the correspondinij 
 
 luction to Differen- 
 
 whence 
 and 
 
 
 d?/ _ 1 
 ^^ V[l— a;'] 
 
 dy~-—J^___ 
 
 
 (568) 
 
 e function, 
 onding direct func- 
 
 V[i~x'] 
 
 (569) 
 
 e ditierential coel'- 
 
 If 
 
 we have 
 
 
 y = tan - X 
 « = tanj/ 
 
 
 
 : '^ II 
 
 i 
 
 f 
 
 
#« 
 
 I ' t 
 
 —148- 
 
 wlience 
 
 If 
 
 we have 
 
 wh^ce 
 
 and 
 
 dj/ 
 
 — 1 + tau- // = 1 -f- «» 
 
 dx 
 
 1 
 
 •l+a;' 
 
 
 d,i = 
 
 dx 
 
 
 U = 
 
 COt_'iC 
 
 
 X — 
 
 cot y/ 
 
 
 dx 
 
 dy ~ 
 
 — (1 + COt* 
 
 .'/) = 
 
 d,j _ 
 
 1 
 
 
 dx 
 
 1+.* 
 
 
 du = 
 
 dx. 
 
 1 + -? 
 
 
 (570, 
 
 (571i 
 
 -a+a-") 
 
 ^572) 
 (573) 
 
 If 
 
 we have 
 
 whence 
 
 and 
 
 y = sec- X 
 
 X —■ sec // 
 
 doc 
 
 — = tan // sec // 
 dy 
 
 = sec // ^[sec* y — 1] 
 
 ^ ^ 1 
 
 dx x^jlx^—l] 
 
 <lil = 
 
 dr. 
 
 •'VL^'-l] 
 
 (575) 
 
—149— 
 
 . il 
 
 -a-^x') 
 
 ^572) 
 (5731 
 
 I] 
 
 (5741 
 (575) 
 
 If 
 
 we have 
 
 1/ = cosec -'a- 
 X = cosec // 
 
 (Ix 
 dy 
 
 — ■ — cot y cosec 
 
 V 
 
 - - cosec // ^/[cosec- y—U 
 
 whence 
 
 and 
 
 dx 
 
 dll z=r 
 
 s,V[a,.«_lj 
 
 fix 
 
 x^lx'—\1 
 Successive Diferentiation of Angular Functions. 
 
 (576) 
 
 (567) 
 
 coefficient ™ay be derived Li I ''"!: ""'•^^^^' « ^"^^^^^^^^ 
 ready explained. This Ht i? ^T^'^^^^ *° the principles al- 
 «econd d?fferentia[ coeffielt?:72 "^''^""' '^ ^'^" "^"^'^ ^^« 
 manner, the second diffe nil 'olffi ''T "' '"°"*'°"- ^° ^^« 
 
 «oon.The notion o.hr^;^:s-:^-:--«^ 
 
 Let 
 
 than 
 
 .'/ =Ax\ 
 
 'in 
 
 dx - ^^'' '^'■«t <l>fl'erpntial cooffici 
 
 d*y _ 
 
 -^^— the second " < 
 
 ent. 
 
 dS, _ 
 -^ - the third 
 
 !M l| 
 
 I' ' .1 
 
 '-■ J 
 
 '1 
 
 
I .'.' 
 
 \ 
 
 -jr.d— 
 
 '''// 
 "7/? 
 
 — Ihc touvtli diliereutial coefficient. 
 
 .md .<o on. 
 
 
 Therefore, it' 
 
 // = win X 
 
 
 dn 
 
 --— = cos .' 
 dx 
 
 
 d 11 
 
 _-r = — sin ;r 
 f /,;••■ 
 
 
 = — cos a 
 dj-' 
 
 
 d'u 
 
 --— = 8111 X 
 
 d\l 
 
 and so on to an iulinitc oide; 
 Similarly, if // = l'Os x 
 
 I'l 
 da: 
 
 — sin X 
 
 ~dx' 
 
 d'u 
 
 :=: COS X 
 
 ~ Sin X 
 
 = cos X 
 
 and so on to an infinite order. 
 
 Tai/lor's f/irorcni and Madauriii'ii thnorem. 
 
 Let ji ■= fvx^ denote M iiinction of .c, av.d /( any finite quantity. 
 Then the liinoniial function /(«• -j- //), when expressed in terms in- 
 
—151— 
 btr ''' ''''''''' P°^^"''« ^' '^ -y be suppo.od to b«. as see. 
 
 Taking the successive difTerential coefficients, we hare 
 
 
 X' 
 
 <te' 
 
 etc. 
 
 ~ 1.2.3i?+... 
 fife. 
 
 t^ar ~ "^ ~ Pf whence /- = ^ 
 
 ilx' *<fei = ^-2^, whence 0= ^■^. ^ 
 
 ^/■oi d';j 
 
 <hf '- ^ 1.2.3^, whence i? =^>_ 1_ 
 &c. ''•*' ^-2.3 
 
 Substituting these values of PO/? ,,, , . 
 
 ^'V,/^... in (^j gives 
 
 ^f'ich is Taylor's theorem. 
 
 (578) 
 
 
 I- 
 
 A 
 
(• '■ 
 
 ■^-»,*,-,-.fts-'J.-'lt 
 
 s 
 
 
 -l.jl'— 
 
 ii 
 
 '.-i^ 
 
 Make .*• — '• iu ,'i7f<', mul 1ft 
 
 X denote wlmt // ln'i'uincs. imilerthis liypotho^is 
 
 A, 
 
 Y II II 
 
 anil so on; we olitain 
 
 ,l.r 
 'I'll 
 
 A.-, 
 
 and substituting ^ for /. in the last expression- 
 
 h" 
 
 1/ = fU'l 
 
 J 
 
 ,+x*+^,-;;;,+x-^;,+ 
 
 wliicU is Maclaurin's theorem. 
 
 Erpamion of An^fular Funrfion^ iu fmnx of thf arc. 
 
 Makin" a. = 0, in 7 = «i>i ^. then // = sin = 0; and the suc- 
 cessive drfferontial cocffici.int of sin x, »s found in page 150hccoiuo 
 
 'III 
 
 ~ cos 
 
 0= 1 
 
 <l.>- 
 
 • - nr SlU = " 
 
 ftl 
 
 'I'll 
 
 = — cos 
 
 = sin <t=0 
 
 and so on to an idiinite order. 
 
 Also making . = in ,v = cos ., then , - cos == 1; and the 
 
 successive differential coeiHcionts of cos ,i; become 
 
 (an 
 
10S18 
 
 -m~ 
 
 '/.'/ 
 
 d'>/ 
 
 - , , = siu = 
 
 d*y 
 
 - , ; cos = 1 
 
 I 
 
 %# 11 
 
 i>. 
 
 r+- 
 
 •2.3 
 
 .•2.3 ^ 
 
 = 0; aud the sue - 
 page ISObocoiuo 
 
 !i«<l SO on tuaij iiilinii,. oi.l.r. 
 
 Substitutiugthcsii values for. r r r v ■ ~ 
 
 -f 1. for tl.o cosim.. f- I, - 1, for th- «:„., un.l _ 1, 
 
 
 T"^4X4.6jIj" "^•- .^^fJ) 
 
 COS .'• :^ 1 ■ I __ ■'■ , .,■'• 
 
 whicllMlv kllO«-,Ml.s/.VVA.svV7Vx 
 
 I'orlMvvity. w,.s|,a]l,vn-U. ' ibr ^ ^ . 1 
 
 (5?li 
 
 1. 2 \:i) "^ X2J' •'^"'^ ''° ""• 
 
 i 
 
 .^, 
 
 5 (1 := 1 ; and tbo 
 
 tan 
 
 sin ./■ 
 cos .<• 
 
 "i3)"^^~--- 
 
 1- ■ -1-^-. 
 
 »"• >> t nin,> thereforo iis.snnif 
 
i 
 
 —]:,[— 
 
 Eqimting thcsfooinl momljcrs oftho liist iwori|u;il,ions, iind clonr- 
 iut,' the I'nictiou, 
 
 = ..-\-A 
 
 (21 
 
 1 ' 
 
 4- 
 
 4) IH-- 
 
 |4) 
 
 Anil equating the coefficients oi' like ti^rnis, woolitnin, after rod ac- 
 tion 
 
 A,= 
 
 •■5) 
 
 Ai= ^-. and so on. 
 
 (5)' 
 
 2j^' 
 
 94 
 
 Thereibre, tan .-• = .. + ^-^ f ^^ -^- _^_ ^ + ... 
 
 i582i 
 
 Dividing (5bl) by |580,| wo aoo that tlie result will bo a series 
 
 commencing by - containing oniy odd powers of ./•, and cacli toriu 
 a; 
 
 having the ngative sign, excepting the first' And proceeding in a 
 
 similar way as before, we obtain 
 
 '•..', 
 
 1 tv 2V 
 
 cota.- = ^ -ro-^-ro 
 
 1. ± 3 1.2. .3. 4. f) 
 
 (593 1 
 
■i:.5. 
 
 .( 
 
 ons, iiml clenr- 
 
 h...l 
 
 tain, after rod uo- 
 
 i582i 
 
 will be a series 
 , and each term 
 1 proceeding in a 
 
 l593i 
 
 K.rpuiiiii.on tif ar<'i in frmi-i n/t/n' sine, rustnc, anil, tangmf. 
 li ij — sin ./■, then ,'• = hih '^ ... Ami a.saunio 
 
 sin-'// ^A-\- //.v4- (y+ /)//'+/';//• + ... (n\ 
 ihirmvntiatiuy both ni> iiibov.s 
 
 Kximndirf,' tlio liist nicinbov of [h\ hy tha Hinomial thoorcm: 
 
 ~^li _^y - 1 +^/''+ o:i ^ 4 27476^' + - <"' 
 
 E(iu:itiii;,' llie coi'tiio(;iits of like jiowcrs in \b) and u-\, wo liavo 
 
 /y=l, C':=() 
 
 ^^^i:t3:4.o'""^- 
 
 and making ij — = in !(?', we have A —0. 
 Subbtituting these valiu^'i in kd, we obfciin 
 
 .(■ = nm^ 
 
 
 (584) 
 
 Since, by the conqilcnicntal relation of sine and cosine, 
 cos-' // = 00'' — sin^"'//, we have I'vidtsntly 
 
 cos- '// .— I- — i.")8ti 
 
 If ;/ = tan :••, then ,c -- tan' ' ;;. And r.'^nunn 
 
 (5851 
 
 i 
 
 ■,^ 
 
■iip-*<i-««MMMfcB*l«MlifeB 
 
 - I .Mi 
 
 t.iii ';/ ~A + it:/ + Cn ). /|v' + ... ,rt, 
 
 T)i(Tt rciili.iliii,^ l)<)tli ii'"Iii1ht.s 
 1 
 
 1 hn 
 Ily tho biuomiul tlu'oiiMi 
 1 
 
 y^-yy + -.'6V/ + ;i %• + ... '(/'I 
 
 !+//• 
 
 = 1 — //' + ,'/' — .// f //" — ... 
 
 Eiiuiitiuf^tlic coiiiliciciits of liki' iiowith of./- in tlu^.-'i^conil iiiciu 
 l)«rn ol' {b) ainl ui, we liuvf 
 
 li = 1, C= 
 
 /) = — !^, A'=0 
 
 Makiu^ // = iii(«l, «inco tlioaHsimKidilowloiicnuint is ttuo for 
 all valu(!s of ,//; thon A = 0. Tluimfoid by Mulwtitiiting tlioso valuos 
 of A, B, C, D, &c., in (rtl, wo ohtaiu, each allomato toriiisi vauish- 
 ing: 
 
 ^ = tau ' ti = // — '.'/■' + I'/ — !.'/' + . . . 
 whici is known as Gref/ori/'s shvIpk 
 
 (58Ci 
 
 Ip'Ii 
 
 Killer's Formuld' and Dr Moivrr'v tormiilit. 
 
 Lot y = c^, whoro e donoton the baso of tho Naporian system of 
 logarithms, then, by (552l wc havo 
 
 .jL. — pit 'L. ^ riX- •!_ — ,,x- Urn 
 
 ax ax dj? 
 
ia> 
 
 •(tl 
 
 .. '.i:\ 
 
 lin HOCOUil llli'lll 
 
 nicnt is ttuo f'ov 
 
 iug tho.si) valuL's 
 
 iito toriu3 Viiniah- 
 
 inHGi 
 
 mialit. 
 
 ^pcriiin system of 
 
 ■ i-u — 
 
 IwlK't', iiiakin^; .>■ <•, wn olitain liy Ml»clalu•iu'■^ tli(ioriiiii 
 
 = 1 + . + 
 
 x" 
 
 1 1.x> 1.2.3 
 
 .SiitiMtimtinK j\/ — i for j- in |ft87) giviM 
 
 + ... 
 
 (587l 
 
 ,.'1 -' ^ If ,«v/— 1- 
 
 1 
 
 •^---^^ /_14- 1' L 
 
 '2 1. 2. S"^ 1. L'. 3. 4 ^ ■ ' 
 
 or, by |581) and AHOi: 
 
 f-*t -' = cos X' 4- V — ' ""' ■'' 
 .Substituting — r for a- in the last equation 
 «- >:►' -' = cos j; — ^ — 1 sin x 
 
 whi'nc(t cos X = Ju''' -' -|-c *r-') 
 
 X' 
 
 1.2.3 
 
 )v-. 
 
 8m u: = 
 
 1 
 
 2V-1 
 which arc Enlor's forniuhi". 
 
 (c*>'-'_ (.-*>'-') 
 
 Multplying (590) by 2, nnd |.')91) by 2 ^— 1 wo obtain 
 2 cos a; = e^t^-' + c-^t'-i 
 
 2^— 1 sin J! = e^t -' — c-r-f-' 
 Dividing (593) by (592) giyea 
 
 e*!'-'— e-*!'-' e^V-i — i 
 
 ^— 1 tan a; = —— = 1 1 
 
 Substituting in (588) and (589) ma; for x, we obtain 
 
 i.J88) 
 (589) 
 (590) 
 
 (59r 
 
 (592) 
 (693) 
 
 (594) 
 
 I '1 
 
 •^ 
 
•;,!■ 
 
 .iiul 
 
 —lOa — 
 
 ,,-„t,/_i _. ,.^,j, ,„^. .^ ^^ — I ^j.j II,, ■ 
 (, ' '"■'■ ' = cos iit.i — ^/ — 1 *;ii '/'■'■ 
 
 lint till' tirat in'!i)ib(M's ol th« two last L-quit-iaai are rcspectivoly 
 equal to :r-''i'- 'i'", and - ''' 'r'", tlicicforr thi' siinii- traiisforiiiii- 
 tioii iiiav Ij'' iii^i'K' on till' SIM . mil innnbu-s: tluit i.< 
 
 'H.*'. 
 
 iiiiil <;o'7 //(J' — -y/ -- 1 ^111 '"•'■ = ''iOf^ ■•'- \/ —"I siiii ..:;■"' 
 whicii writtini in oiu; i'i|uatiuii, bi'i'uiiii' 
 
 cos iiij:-\- ^/ — 1 .sill /«.i: -■ icos-j- ^^ — 1 sin .f '" loO'i 
 
 ,,i'..', 
 
 which is !>(' Moivvc's t'onimi/.. ainl i^ tvw I'of nil iiilctiral values of 
 
 IH. 
 
 To obtain a cdiniilctc fomi oi l>i' Moiviv'- iuviiiula, «;■ havn urM 
 j)ly to suhstitntc .v-f •_*(/•:, for .c, Avliiiiv k dciiott'sany inieyral uuiii- 
 bur, lioiicc 
 
 ICOS ./■-!- ^ — 1 sill .'• '" ■-■- COS /.'U'-j Jil'/z/rl 
 
 -j- ^— 1 sill \iiij--\- 2aii>--\ (.")i)G' 
 
 which i.; true tuv ail vaiuos of //-, iiitc;.;ial or fractional. 
 
 ]jy tho ins])cctioii of L'c iMoivrc'.s formula, wti .sw' that llie uxiiic, 
 sioii co.'s .<; -f- ^/ — 1 .sill ./■ is raised ;o any ]'owor by tho Ti;ultii<lic, 
 tion of the anyle 
 
 4 , , I 
 
 Ji.rjiaiixiiiii .' j,ijin'r.<i ('/sill ,c fi/iil cCS 
 
 Let 2 = e*^ ' , tlieu wc have 
 
 (.""^1 - ' = ,;"; e- ^1 ' r= 2 ' ; and c "•' i ' — .r ''' 
 
 V cos ./• =: ;: 
 
 •1 ,J ~ 1 sill .,• 
 
 From I)e Mci'Te'? foniiula \»e deiliio«' 
 
 (en 
 
-ir,9- 
 
 '' cos /ij: 
 
 ■■/I _i- •• " 
 
 are rcspectivrlv 
 ■amc traiisforiiia 
 
 1 sin .r"" 
 I siji .. ,"' 
 
 inlc.gial vuliU'.'. of 
 
 li:i, \\- have n'.m 
 .iiy inti\L;ial ir.iiii- 
 
 -|- 2«;/' vi ,.'''.•0 
 [ictioiml. 
 
 ;l' that the U.\plrs- 
 
 liy thfwiiultiplita- 
 
 IS .(,•. 
 
 -■V' — 1 f*iii /(.'■ = ,v" — .V " ,,/| 
 Also, .,-o liave l,y raisiii- i,m to thc^ /,th j.owor 
 ■2" eos".c = ,^-L^ I ,( 
 
 r.ut 
 
 ' — ■•- -f- I'.:" '^ -I ..ti-^i I 
 
 whicI,bc.co,aoM,yeon.hinington,.:at o,,,.! ,,i.tancos iVon, ti,o 
 
 , IH71 1) 
 
 that is 2" cos 'V =r 2 
 whptiCG dividing by 2", 
 
 cos 'V =2po.s,,r + „r2 co,s («_ 2., -I 4- -""" ^ ' 
 
 -^ 1.2 
 
 [2 cos in— 4,.-] -J. 
 
 cos ".,• = __ _^cos «,,• 4- ;, cos ,;, _ 2 .,• + n']^ }}_ 
 
 1.2 
 cos (n — 4v;+...] i,-,n7; 
 
 Since in the cxpnn.sion of 
 
 r i2+^-'i" there lira w_l terms, it 
 
 ^^n.t.hen.isoda,,.+i;.i„i:;e;,:r;i.:^^^^ 
 
 no to >s ,ndep.n ent of,, in ,597.. If on the contrary „ i, .,,„ 
 
 ^ in 1 ' ' T',*^^'"' ""'-^^ '" ^ -i'ld^'" t-mindc,H.ndcn 
 
 01 z in the seru.s ,.., wliich will he of the form 
 
 I 
 
 1.2.3.. ~{hi —•m« 
 
"'^^iima. 
 
 
 ^■J''. 
 
 — 100— 
 
 iiiid tliciijforc ii twill in ir)!)7| must lie fouiul iiidcin-'iulont of .<•. 
 
 Kxuiiii)lc 1. cus' .'• = - , icus ;">,/• -|- 5 CUM ,V + lit cos .<■ 
 
 2. cos^ r = -- .cos 4.C + 4 cos Ic + Gi 
 ill ;i similar way, raisiiiii' k/i Io llir cth jiuwi'i. we uLtaiu 
 
 O?! 
 
 "sin".' = i,v— -:-' " 
 
 Hin — h . 
 
 4- 
 
 l.-I 
 
 -01-1) + ;, I ,,.| '»— .) 
 
 15981 
 
 (399) 
 
 (600) 
 
 when n is even. 15981 hccoiiios 
 
 ^n^^ — [(» sin".''= I.:" + z""'-- 11 '2"'~"' H- «"*""*' + 
 
 whpii n is odd , (o98l Iwoiiies 
 
 •2», ^ — li" sin ".'• = I.;" — .•.■-"! — niz"-"- + z^!»-'h 4- • 
 
 "NVhonco we soc tliiit in (lu' liiat case, the hiiioniiiil iz+ 21 liav- 
 ing the sign +, the series will involve only the cosines of multi- 
 ides of.'' :md in the second case the binomial having the sign — 
 the series will involve only the sines of multiples of ./;; see la) and 
 I//1, i.a.jc 158. In hoth cases, dividing hy 2^\^ — l\n^i\\<ri\G the 
 expansion of sin ".'•. 
 
 Example 1. To expand sinV. we have, hy (600) 
 
L'udont of .»'. 
 10 cos ,/■ 
 
 ■ (J) 
 
 (■ uLtaiu 
 
 -,,■1 '"—■■' 
 
 -.'•!-"... (SOS! 
 
 -')i + ... (3991 
 
 ■') +• ... (600) 
 
 lial iz-{-z\ hav- 
 losines of multi- 
 ving tlic sigu — 
 of .c; ace (a) and 
 \]n Tf'illgive tlio 
 
 0) 
 
 — Ifll- 
 
 /-Irmn',-.- ^,/-ism3.._ 
 
 6\ — 1 4n.r 
 
 \i .'.'•Uci' 
 
 V -- J ■• sill- 
 
 Mil .-— .^^. sin;;..-- 3sin.?-i 
 ^••-"PloO. To..x,.iadsur.,..ohave,by.;.99, 
 
 •J cos 4,- ~ s co.^ !,■ 12 
 1 
 
 ^, cos 4.,-_4 COS 2.C + fi; 
 
 wlieiicc 
 
 siir.'' = 
 
 cos .,• -,- 
 
 an 
 
 1 sin.,. <. = co.s;„,-L , . 
 
 . A — - J sill // .r 
 \/ — I sin /I.I- 
 
 "•ll.'IKV -■OS ,;,,. ..= ,,0, u. / , . 
 
 '■us //..■ ^iCOs ,,■__,__ I ,• „ A/ — I .SlU «,,• 1^, 
 
 e , V — i sin a;)"! 
 
 "'"'""""'■''■■" """.>■"«"«.,„„,„„„,,,_, 
 
 sill /(.'• — 
 
 icos,/ ~^/i sin a;)" 
 
 ^''''^""""-'"^'^^'^'■i.s.wcol.tain 
 
 cos /,.,• — COS",' "'"— ll 
 
 + 
 
 '""l-lM«- ,„__. 
 
 / 
 
 1^ 
 
 
 1 2. 
 
 . 4. 
 
 cos »-4,x. sin4. 
 
■^•Wk.!. 
 
 r-'iriiriiir 1 1 iiiiwiiiiiiiiiifB— ■ 
 
 [■f' 
 
 \ 
 
 — l&Jt— 
 
 nn- 
 
 — 2 m — 3i 1)' —+) i/»— r>i 
 
 1 1 i/( — i! \n 
 
 1. 2. ;5. i. •'■'. 
 
 .. eo8 " '' .'• sill "n 
 n{>i — \} (!»■— 2i 
 
 -f ... filJli 
 
 gin n* = »• cos" '-r sin .»• — ^ .^, .^ 
 
 X cos" J- siu- 
 
 ■ :',ti 
 4 I ■■ I ' 
 
 ii;nj— 1) Ut — -21 I" — 3) !» — 4) 
 
 < Ros " •'•.»■ •fill X 
 
 run— h [71 — 2l \n — 3) (n — 4i i« —5. /( -«■ 
 i. I', 3. 4. ii. 6. 7 
 
 . pos" V sin ./■"4- ••• ^0- 
 
 Reinlutioii (//sin J' '(»r/ oos .c hit(>f<i<l»rs. 
 
 The yalue» (», -, 2r. 3r,otc.., make sin ny - r n-. ihoniforo by the 
 Theory of EquatioDB ^re Ahjrhrv we Ijuv.' 
 
 iin j; = '»4^-^-] •< [.-•--■^-r] - b/---^-''] •■■ 
 Pividing both membors by x, ;iii.l dfcomiiosia',; iMch fuclor in thi' 
 second mnmbpr. we obtiiin 
 
 1-=. .=(>-^^j. ..=(.-,:=) ■-.^(>„:l) ... 
 
 But if wo niako -■ = (», tlie tlvst iiu'mbfv - 1. thi-tct'on.' ihc svc 
 ond member mast also be unity iind wr li:>v.^ 
 iir- Jr ■ .ir -.. . — I 
 
 Hence ainr = j, i 1 — ^, | y 
 
 i).. 
 
 .fiuS 
 
 .Similarly, tin' viiliies of ^r:. Ir. \-r. mukf cosy - n. lli^iuu 
 
 OOfl r = /([/•! — . ^- -J ■: [■'•^ — ' i- '] [■'■- - i" " ] • ■ 
 
 X^ 
 
-Hi3 - 
 
 4l \n— r>) 
 
 r.siii"n-|- ... 6<'l. 
 
 — ' 
 
 ■2\ 
 
 
 
 :t. 
 
 
 
 
 O." ' 
 
 
 ' siu''- . 
 
 ;' 
 
 ■ 4i 
 
 
 
 
 w.< 
 
 " ■>. 
 
 1; siu X 
 
 
 ■5.1 
 
 i/( 
 
 -C 
 
 
 ■ sill .1' 
 
 + ... 
 
 602 
 
 /<" 
 
 7-y/V 
 
 <:. 
 
 
 II: 
 
 tllP 
 
 I'oforo 1 
 
 by the 
 
 '-' _ 
 
 -,;',; 
 
 .r]... 
 
 
 a'.; 
 
 ^'lW. 
 
 K frtctoi 
 
 ■ill tlif 
 
 ir- 
 
 ■(' 
 
 
 ,)■■■ 
 
 (' 
 
 - 
 
 .:;.)■ 
 
 
 I 
 
 lint if wp luiiKo.r = 0, thoucoH.'- = hand thoroforo the second 
 iiu'iiihcr must who he (-((ual to unity, and \vc have 
 
 Tlici'ot'ofe 
 
 co.s .(• 
 
 -(-?^)('-£)(>-^9... 
 
 604) 
 
 CompHtatiun of -. 
 
 The numerical value of r. tliu ratio of the circumference of a cir- 
 cle to its diameter, is most easily comiaited by means of Gregory's 
 series 1 ;"i8fi 1, 
 
 \Ve know that tan 4.-)° = tan }r ,= 1. Therefore making x = 1 
 in I ")8() I we oliiain 
 
 ■ , .i^=l-i\4-l-! + i-... (005, 
 
 l!ui thus series converges too .slowly tu be of any use. 
 S ■! i- ■■ more rai)idly converging aiv obtained by resolving i" into 
 
 two (.1- more ar.;.s whose tmrgents are known, and developing each 
 
 uf tlies,- ares li.v (Iiegory's series and taking the algebraic sum of 
 
 IIh' I'esiilts. T .us if we assume 
 
 1. tiii-refore the swc 
 
 we have 
 
 JT = tan '//( -f tau-'w 
 
 (606) 
 
 If 
 
 I 
 
 :'|... 
 
 ■ Cos 
 
 II. llt^iee 
 
 whelic 
 
 or 
 
 tan JT.- 1 
 
 _ (/( -f" // 
 1 — nil/ 
 
 1—7/. 
 
 Ill — ,«) 
 
 l-i->l 
 
 1 — 
 
 I + wi 
 
— IGl- 
 
 Tlierefore giving any value to/;', ilic cunvsjioiuling viiluo oi n 
 is efisily found; or couvi'isi-ly. 
 
 If we take rii = ^, we tiuil liy h .k = Ij; whence by tii'ii we havt 
 
 ^- = Uin '{, -i- tiii-^ 
 But by I J regory's series 
 
 =z I), 0(1(101(11(10 
 
 -4 o.ooo:.''ii!ii'! 
 --- i>.("iO„'l7ol , 
 4-0.0 :oo(H);j'.i , 
 
 -I-- O.OdlOOOU . 
 -j- 0.000001 (0_' . 
 
 — 0.. ")()(; tjiiso . 
 
 -- 0.4(J3(i,76i; . 
 1 I il 
 
 --().04l(;(i(i(;6 ... 
 
 — 0.(1(1111(11)7... 
 
 — 0.0(1' M)M.;j,s ... 
 
 — 0,000002(1.". .. 
 
 — o.Ooooooio ... 
 
 — (I.O(UIOOOO() ... 
 
 — o.()4-28:i02t .. 
 
 '--'>i-io+io- ;o+- 
 
 -p '. .OO''-".:;!! ! 
 -f O-ooooo^iM 
 -f 0.1(101100(14 
 
 = 0.334 1620."), 
 
 — O.lllL'.U .71 ... 
 
 . — O.0oo;i(;ri;i'-' ... 
 
 — o,00('ooO"i! ... 
 
 — o.ooooooou ... 
 
 — O.Ol24llo7 ... 
 
 tharefore 
 
 = o.:);M7.5ol,-i ... 
 ^T = 4U;i(il:7(L' ... + o.3J17,")(i4,'S ... 
 
 - 0.7S.".;i'.)Slo... 
 
 ^hfnce r = 3, [1 1.VJ2.40 lacun.tely to .si.x places 
 
 It may be lakcu as a general iiilc ili.n. in computing an approxim- 
 ate value by means of intinite scii.s. wf must e.xteml each term of 
 the series lo twojilaee.-? olMeeinials uljic- than the n(oulir; of deci- 
 mals intended in' the linai ic-uh, and laki tiie t.inis met with be 
 
-ir.;-)- 
 
 V 
 
 11'' v:ilui' of w 
 
 O(H') wt^Uiivi 
 
 ltif)(;fin. 
 iiu;ii7. 
 
 imonli) . 
 
 4HI(II)(I0 . 
 •>82'J2-t , 
 
 for<' n'lch'n" siicli a njic us hiixii!" lor its first lU'CumiLs as 
 
 many ci- 
 
 jilu'i's as tlit'ii' AW. piaciift ul ili'ciiual.s in imjIi tciiu. TIr'U tlie last 
 two ilccinials of lliu result liciug llir only oiifs wiiicli an.', in uU cases 
 possibly liable to bo all'octt'd by tln' '.'jiui' icriuf neglected in the 
 series, are rejected, and the utlur-i assumed as accurate. 
 The loUowiuL' is a more ranidlv coinn'i'inL;' series. 
 
 \V 
 
 e rave 
 
 tan 
 
 tan- 
 
 Ian 
 
 tun 
 
 1 — tan 
 
 tan 
 
 ■41) 
 
 - tan 
 
 - 'an 
 
 tai 
 
 - tan 
 
 .-.^ tail 
 
 tan 
 
 :;<!) 
 
 U.Tl ... 
 ii'HiDrrJ ... 
 
 iiuO'i! ... 
 MllllKIU ... 
 
 111157 ... 
 
 Adding the above eiiuali 
 
 (uaiionn: 
 
 tan '1 - 4 tan- 
 
 tan 
 
 '(-Jo) — 
 
 Wl 
 
 lonce 
 
 .h-= It; 
 
 tir , — tan 
 
 li.T.t 
 
 •230 
 
 :(;07l 
 
 whicii is known as Much 
 
 Ills serir 
 
 Now by Gregory's series, tak 
 
 result: 
 
 iug jight decimals to keep six in the 
 
 l7.")()l>S ... 
 
 tan 
 
 1 1 
 
 ;V .) 
 
 '(:)+!(:)";(;)+ 
 
 to six places 
 
 an ajiproxim- 
 
 ll each term of 
 
 liiubr; of deci- 
 
 luLl with be 
 
 -■= U.L'OdllOOdll 
 4- i).()0(i(in4()O 
 — - (t.UI Mil III! )().-) . 
 
 = it.20(Miri-t().-i , 
 
 n.ll)739."i.3i 
 
 o.o()2(;(!f)r)r) . 
 
 . (I.(i0i)(ini82 
 (».noi)( 1110(10 , 
 
 ■O.0o-2()'j,><-18 , 
 
'-.^-..Kt^,^^Z.:^iu'J)N^^>m 
 
 
 m 
 
 'i ,1 
 
 _-ir,f.— 
 
 wlienrr -Itim" 
 
 1 
 
 AU 
 
 \M\ 
 
 ' '(-V+- 
 
 Thovefori' 
 
 1, _ (i.;s'.ir)8-Jl8, 
 
 .'.sn 
 
 = 0.(:('41S-Hn...-0 0()nOOOO: 
 
 = 0.0(U1S403... 
 
 _, (i.(i(illf<tn;'. 
 
 0.78f):VJ8l.")... 
 svhence rejecting, the last two .l.n.n.ls: 
 
 Natural Sin. and Cos.n. ^^^-^-^^l;;] '^Z;::!.^.., 
 
 the nunilx'V ol tiMin>tt"H i 
 
 iiiteniloil. 
 
 Thustolinacosin\Nv.'li^'v.. 
 
 Suhstitutiu,tlusv.lm ior.in,:^sl -obt.in 
 
 l(r = 1 OOOO(KHH) _(l.(>l.V.':.ll><i... 
 
 "" . 4(,,oooo:'sf,C,...-iMiniKi(iOol... 
 = i.()n()():^S(;(i...-<M)i.V23(i'.)(>... 
 
 = ().()84Sii77('> ... 
 
 . .■ „ , l.,.i two .UnnuKilH .s iM'ins.- linMc ta !»• altero-l Lv 
 and reu-ctiug t^'^ -'J'^ ' ' ,^,^^..., ,,,,„„,,,.,, ,., six place. 
 thrlifthtevn.m^tho^on..^w.^. 
 
 Anothov n.Ll .oun-l l.y ^-uLv, H.^ inv->tov of iW two s.u. 
 ,,X.ntiun.l...onsis,sotn.Uin,,,n.lu..,s.n,.s. 
 
 •'•= '"•^" i. 1-1.1 
 
 „.t-mt coi'tlicionts 1)V nicansoi wliicli Ww 
 ■ind then of comimtuig I'on.lant .ouncRiu > 
 
 ,\na men calculated. 
 
 111 i;^»t'! anu '■>- ^ < 
 
10 
 
 ■iix 
 
 lilUf^i'-* 
 
 
 /,',. 
 
 
 
 
 i v 
 
 ;'.,. 
 
 ■■(•rii - 
 
 .".811 
 
 ;\ni 
 
 ^ then suiuuini;- 
 
 v u 
 
 ■ ill' 
 
 ■ilUill 
 
 lace 
 
 Km- 
 
 lo},' rtill IliAr: - loy \k -f li;j,' Hi -f log 1 1 - - "'^ i 
 
 D-'vlopiiiy i.lu.sr lu^'iirithiiKs Mil tnkiiig the .wnition aystoni, 
 when luodulu.- is .IfuotM.! l,y M, w." Imvrt, arr'i.-inf; iK'conlin;,' to 
 the powtMs ot ;/' 
 
 lo;,' sill m.^T = 1li'_' .Ir -f- loy 
 
 ?/'. 
 
 — m 
 
 ■"(i+7+^+.) 
 
 
 w 
 
 -^(i-xL + i , \ 
 
 •J 
 -' II 
 
 \' 
 
 f 
 
 I 
 
 ' W 
 
 IC"' COS ((/.J 
 
 A.:..U/(l+-+.^.4-...) 
 
■ .t . amiiimi tm 
 
 ^1 
 
 \ 
 
 **■•'.': 
 
 'IS 
 
 I'llicli. ;it't.|' tlics:iliilH!iti(.iIl III' lli(' coil-lMnt III 
 
 iiiiM'iii'al s(U'i('s, II 
 
 ml til. 
 
 sulistitiiliMii mI I'll' V'lliv s ul' .'/ iiiiil }.-, ii'wi' siM'iiN proj,'!' 
 i;ovtlinu im iiu' asrriiilii 
 
 OS.iiU'' 110- 
 
 ix vvwt'V'- oi III. ric'i icuii coiiriiiniiiL: 
 
 '•VU- 
 
 stiiiil lai ri)V 
 
 llll - tlir Ml')' ( -^ III' 1 ,11' liniiiivical larl; 
 
 mi" slowly ooii\('ig'iiL.'. 
 -.lii'V lie ni:iil'' iiii'H' iiiiiilly cuini'r;.'!'!!;.' Iiy kt'<'i>iii>,' sonii'ol' tlic tii>t 
 ti'riii> ill till' |ii'i iTi'ding roriiiuhi 
 
 It is rVillrlll tllMl lllf i^VlMtl'V I llC limilliCl' ot' tcrills VI'tailKMl, lllc 
 
 •null' raniilly cuii cr^iii.i;' will Ih' the serio-i tbv the other Icfiius- 
 
 '(,u iiiu:f i'irti'-ului~ wr ivl'i'r I lie >ln(lciit to tiic [)ivi'ac'c to (.'alll■t'^ 
 
 Tal.li-. 
 
 ti 
 
« " 
 
 PVKT II. 
 
 SPHEKH'AL TKK^ONOMF.Tin'. 
 
 OlUEf'T OK SpIIKKIi'AI. TMOO.NOMIiTUY 
 
 8PHBHI0AL Tbigonomkthy tivatji oi'lhi' niwlhods ol determining 
 the several pwts of ii spherical tiianglo_^lroiii iiiiving coit^iin pnrts 
 given. 
 
 A Spherical Triati'jle is that jioilion oflhi/ siiiikue of tlie sphere 
 bounded by arcs of lliroe gi-eat ciioli^s; that is, of thiee circles 
 whoso phnes pass thiough ihe coiitro of the sphere. 
 
 We here suppose ilie student to be ac4Uiiinti.'d with spherical Ge- 
 omotiy, at least, of so much of it as is fouud in Loomis' Geometry, 
 Book IX. 
 
 In the following i>age8, we confine our attention to such Irianj^los 
 only as iire treated of in Geometry; that is, the mialler of the two 
 parts into which thi- surface of a sphere is divided by the three smaller 
 arcs joining, by pairs, thre points on that surface, and which 
 are not in the same great circle. It is evident that those triangles 
 have every one of thfiiv parts le.ss than 180'. The chief reason is 
 this limitation is that, while simplifying the theory, it excludes "no 
 arc or angle which is over nrcessaiy to consider in the practical ap- 
 plication of Spherical Trigonometry. 
 
 
 ■I 
 
r ) 
 
 I iiAnii; I. 
 
 Ui.;i.ATioN> iiKiwiKs rni: Sim ^ vm. iiik An<m hh ok Si'HtKU*i, 
 
 I'lllVMil-IIN. 
 
 Fuiulnnisiiul I'ropoiiiiuti :iml J<'tMiiml«. 
 
 /„ ,( Sfhfrir.al l'n-\n<jl'; the roniw of any side is equal to the 
 r>,utinut,i i-ru:lu>-( „f the ro.iine of tin nitpusite nngle and the tine 
 ui t/ir nthr ^kI'", tu,i>th r irifh thf jiioilwJ of the eo!*inr.i •/' these 
 
 titles. 
 
 rhJM piopo^'iliuu in itiilU'il iuudiuueutul Iwcavwe it serves tw a 
 
 kny to ill! ilif ivM. \Vf ^hull lii«i estiiblish if fiorn thu imspwction 
 uf llic iinu.xK'l li ,'uiv, l>.v inraiii of tk' pnacii)le8 ef I'lane Trigo- 
 uoiuetiy, an.l, lu^xt, wf shall i.iocoed in JoduciuK (tnulyticaily th» 
 ijiiwt iiupoiimi n^liitions oxishnjjc Jietwi'en Angular lunotiona in 
 a vSphurii'al TriiUi^lu. 
 
 L,.i .4//r lie a 
 Siiht-r'uMl Tiiiii- 
 
 ti'ii of thi> SjiliiMi . 
 
 Ml'IlOtt' tk MU-^ll'S 
 
 l.y A. li. '■: iiinl 
 
 the wiilt's op|iositi 
 
 to thfin rcs|ir(t 
 
 iv«ly l)y '(,/'.'•. At 
 
 the poiut A ilnw 
 
 ^rtangeut to the 
 
 arc AB, au'l AT 
 
 tangent to thf luv 
 
 AC. Thf.n hy Gooimlry (Loon.is. Book IX, Vr. VI li the angU- A of 
 
 the Spherical Triiuiiil.' is measured by the plane angle TA T. — Join 
 
 OB and OC. I'loduce OB to uK^et T, and OC? to meet T; then 
 
I — 
 
 Hi* f,l 
 
 I OK Sl'UUUCAt, 
 
 gk and the line 
 eoainrn »f ^Ae** 
 
 iiue il selves am a 
 )m thii inspMiion 
 M b1' riane Trigo- 
 >i; analytically the 
 zulttr I'unotions in 
 
 Jtiticc thr radiuH of the S|i|i(vc is ulso llin liulius ol' the eircleH to 
 whicli liuiou'^ llu' arcs .1/.' uiui AC, oj' in tli« Mi'CiUit ot Ali; and 
 1)/' tht) Hccuiii {ji AC; llmt i.s, mukiiii; ilm radius iKiual to unity: 
 
 t)/ 
 
 ■- .sec A( = H((' /. 
 
 JT 
 
 - xec Alt — Mu: r 
 
 A J 
 
 = tan At' = tan /, 
 
 AT 
 
 - tan .1/; . tan - 
 
 Joinini; '/'/'.wc Im^*. Iiuiii (ilaui' ttiatigk's TO J', and TA'I', by 
 i4S7i ri. Trig. 
 
 yy =^ OJ •■-; or— L'or x wi x coa Jor' 
 
 = .scu /' H sec'e — -J »ec /- si'c i- cos ti [a] 
 
 /"/• ^Ar+A'/'—'2A'r .AT < cos TAT' 
 
 ~ tan // -( t;in' >■ — 2 tiin /* tan r cos A [b] 
 
 Uyil'Ji Pl.Trig. ihc iMiuation [a] becomes 
 
 T'l -T 2 -j- tan- /. -• tan' c — « sec i sec c uos o [c] 
 
 Kqurttiui,' ( b] and [-■]. ii'jcctin;,' tin- common quantities, transpos- 
 ing, und tlividin;^' Ity 2. 
 
 sec b nee c cos <i = tau // tan -• cos A -\- I 
 
 Or, bvi20iandillil'l. Trig. 
 
 cos a sin b sin c cos i4 
 
 cos h cos( 
 
 cos 6 cos c 
 
 + 1 
 
 ^ 11 
 
 Ivm tbe anglt' ii of 
 Ingle T.4 '/•'. — Join 
 It to meet T; then 
 
 Multiplying by co.s /< eos (•.• 
 
 coa (i = cos A sin 6 sin c -f- cos b cos t 
 Himilarly cos /) = coa B sin <i sin c 4 cos 
 
 CO.S c = cos 6' sin c. sin 6 + cos 
 which express the fundamental proposition sis above enunciated, 
 
 i b cos (! ") 
 
 (I cos c r 
 
 i « cos fc J 
 
 |1) 
 
^'^.^^^-'..x^, -iBaigiia'Js^a^a:^^ga 
 
 h »' ■ 
 
 — 172- 
 
 I i. 'r' 
 
 From (li we obtnin evidciulv 
 
 sin /■ sill ,• 
 
 CoH A 
 
 Cos 7j' = 
 
 cos /) -- I'us (^ COS c 
 
 ■siD r/ siu (■ 
 
 Cos C = 
 
 COS r — Cos a coa /< 
 -iu (J siu // 
 
 (2) 
 
 which e.cpri'ns the rtt.sinc of I'lvl, (in,//,' ;„ /rfm.-- i,/ the .-iines an'i 
 cosinex of the sides. 
 
 Subtracting the first loniiuk ol' i2' from, mid addini^ it to, unity 
 we obtain 
 
 1 -|- cos A 
 
 cos (I - COS b cos c -t- siu b ain c 
 sin l> siu c 
 
 '.\' ■ i 
 
 cos II cos l/' 4- (■) 
 
 sill // sin c 
 
 ■2 mi I w,-f /, -f r] sin j> (b-^c — ii\ 
 sin /) sin (■ 
 iiiiiking^ii -•:= A Kj -- /) -f ,'i: 
 
 _ 2 siu wlsiu !«-- »{ 
 sin /y siu c 
 By a similar process we obt^iiu 
 
 1 — cos A = 
 Multiplying [d] by [c] 
 
 ■J sin i.v — h\ sin ls — 
 
 US' — C) 
 
 siu /; sin c 
 
 It^l 
 
 w 
 
 "' ifi. 
 
-173- 
 
 #11 
 
 ¥4 II 
 
 the nines (in<( 
 
 linjj it to, unity 
 
 cos- A = 
 
 Bin ,v sin i,v 
 
 sin i,<( — ^. 
 
 whence, hy [</] j.age K 
 
 *iii-' 6 sin' 
 
 iin (*— c)^ 
 
 sin ,-1 = 
 
 _^^[ 
 
 wn ,N sin ,»•—<,,! sin |^_^i 
 
 Similarly 
 
 ■<in // 
 
 sin,A'_e) 
 
 sin c 
 
 sin 
 
 D 
 
 (3) 
 
 n _ - V L--^'" •>■ sin ,«_„ I ^n ,^,_j, gj^^ _^ 
 
 (* — c) 
 
 ■'in c< sill 
 
 which 
 sldeg. 
 
 sin 6' = 
 
 f'a7.rftv.y //„. .,//,,, ,,, 
 
 __ - \ L'"*"'--sin(.s — 
 
 "sin;,^-/, eijjis— c) 
 
 «in ^- sin I, 
 
 i:ii< li am 
 
 ''■ '"'erw, ofth.,;ne,o/ 
 
 the 
 
 From [,1] wo obt 
 
 un hy ;38Si I'l. T 
 
 rig. 
 
 cos A ^ ::; 
 
 cos -.'> Ji 
 
 Ji Mil .s- sin .,v - a \ 
 
 Ji 
 
 *'" ^ si ,,v _ /, 
 «in (V sin »,• 
 
 ) 
 
 cos 
 
 ^V( 
 
 14) 
 
 ^ ' sin ./ sin h f I 
 which ej-^^rew /-/,<■ ,.,;,;,,„ /• ; . "^ 
 
 From [ej we obtain hy ,357, PI. Tri,.. 
 
 sin } A =^ ll ^'° '•"' - ^i sjn is -. ,., \ 
 ^ ^ sin h sirx:- ) 
 
 sin A h = //si" '« — <i) sin (s~ t-; i | 
 ^ sin a 8in7 / }■ 
 
 in i (7= //«iV^--;^) sin ,,•< — 6, 
 sin a sin f^ 
 
 (5) 
 
••pi" 
 
 ^* 
 
 — 17-1— 
 
 which erprfus the Sin/" uf Itulf nf I'Ofh arKjlf in tcrmx of flu' Sinea 
 (if thf giih's. 
 
 IMvidiiij; ifi' by i4' gives 
 
 tan Ji A 
 
 tan h B 
 
 
 iiu .1 — ht sin A' 
 sin »■ sin ».-■ — '(I 
 
 sin i.s — It' sill us — ci 
 
 sill .V sill ,«,•— /)■ 
 
 ) 
 
 (61 
 
 , , // sin !,>• — a\ sin 's — hi i 
 
 Uni '■ =J\- — : . I 
 
 V \ sill .V sr.i i,s — CI / 
 
 which ej:pre!iS the tangtut vt half of gash amjle in terms of the Sineg 
 of the sidi'.i. 
 
 Dividing ^i iiy i5i gives. 
 
 |7) 
 
 coti^:= Ji-^^'^i.^'^i^-^LJ 
 
 V \nh\ i,v — //I sin IS — CI ' 
 
 cotiiy= /(-^^^'"^'•^M''--^'. \ 
 
 V ' sill i.s — ai sin [s — c ' 
 
 v\sin'».' — a I sin us- — bi f 
 
 which expffn.^ th>- Votanijcnt <it hal( of each amjle in termx of the 
 sineu of the tules. 
 
 Let the polar triangle ot ABC lie .1 B'C , where the angles and 
 the sides are designated respectively by ^4', B', C and a', b' .■'. Then 
 since the turmul.e abuvt; established are tiiie for any spherical trian- 
 gle, we hare by i- 
 
 CCS /C' = 
 
 cos a — cos // cos c 
 sin b' sin c' 
 
■J.* 
 
 in.1 "/ t/ii' .S'/if'x 
 
 tnns 11/ the Sinea 
 
 .17.-. 
 
 r.iii iiv ttc'iiKiiv. r.oLiiiis lu.k tx. f'juf. IX. I 
 
 I*- 
 
 fr = 180° — i>. 
 
 I" -^ im' — r, 
 
 ,/ - 180' ■ .< 
 
 ,;v' = 180" -- a 
 
 'riuii'oforo 
 
 __ ._ eos(liSO=' — .4:- -c(w ac't'^.-iyi 008(180" — C 
 COS . ( 11 -= - ^^ ri80"-'"-"/;i siuTl80'^'^CJ 
 
 which, by 8.5) and i^fJl PI. Ti-i{^., becomes, changing signs ia both 
 members, 
 
 .SiiiKl;n!y 
 
 ('OS (^ -- 
 
 <v 
 
 ■>in /)' -JJH (' 
 
 CDA It -^ I'U.S A CO* ' ' 
 
 .si» A sin <' 
 
 ifo-t (■'-'- '^o.s .4 ci's /»' 
 sin .4 sin A' 
 
 i8) 
 
 //» termx of the 
 
 which exj^fPiix ilir I'lmm nf cwh si'lr >'n <*'/•.,,','> "/ ihr yini'J< nuU ro' 
 ^inei iif Hi': ,iiitj/''f. 
 
 \\\t Hc.o by llie iibovc ilciliiciiou lh;i', i.ny tVinmla oi'a spherical 
 Iriiinj^lo )iiay be, by nitMiis of liie pohtc triangle, tninsformed into an- 
 other, in which angh^s l;ike the j)liu;(' of sides, or conversely. Thus 
 till! f'ormiilif (I:, (3|, (4i, '» . i(> ^'ivc vt'siicctively the following, 
 
 wiit'ii ,s -- i A 4- yy-f- r': 
 
 008 A = COS a sin 5 sin C — cos B co« C 
 cos a = COS b sin A sin C - cos A cos 
 
 
 cos (' = COS c sin ^ sin B — cob A oos Ji) 
 which lupress the cotine of each angle in terms of ih« opporiio sida 
 und the other angle*. 
 
176- 
 
 siu a — 
 
 •J ^/[_ COM S cos •>' — A cos S--JMco» >S ■-( 'i ^ 
 sin /i sin '' \ 
 
 siu /' 
 
 i;^[ — cos 6' COS S A C09 uS— B cos •'»' Tl^ '. 
 
 sin .4 in C 
 
 '■ , lOl 
 
 SIU '• = 
 
 v/ 
 
 [--CO»*'(!OS .S „.| ,;us iS— /i cos ..S' -Ol I 
 
 : 111 ,1 siu /)' 
 
 lit' fill' iiii.iilr^. 
 
 COS \ H -- yjy 
 
 •OS .>! 11) COS N- -C 
 sill /)' sin (' 
 
 ) 
 
 COB 
 
 _ ./COS i.s ^1 cos I ^ — rv 
 
 - ' ' \'\ wn^s.-n'- / 
 
 //(:08 N .(' COS ..S- /)' \ 
 
 ^ '■ '''' v' sill A sin /; / 
 
 (lit 
 
 which M/irvif fill' ■•iisiiii> nf' Jinh' of i^u -ji siili' in fi'riiia d/ flii' siiifn 
 mill coxitii'y "f thi> nu //r.i. 
 
 1 1 cos S cos .>' A \ 
 
 sin i, II -- Jl .,,■,• I 
 
 \ ^ sin /i sin ' ' 
 
 .nlli = Ji-'^''-', ") 
 V ' Sill A sill (' 
 
 8111 J, ' 
 
 I _ /, -COS Xcos >' • C)i \ 
 V\ sill A sin A ' J 
 
 which i'/-/ires.i I he ninr of lni'.i af nir.li sidi> in terma "f tin' aims and 
 cosim'ii of fill' nni/lns. 
 
 X ^ 
 
T**" 'WP 
 
 -fei^ 
 
 ir itiic..- ,(ii'l I'DK'ine^ 
 
 fiTiim of fiti' si Ufa 
 
 rms '/ till' tii)irs find 
 
 tan J „ = // - -Zri^^J? cos i,S|^~ ^ i v 
 
 V \cos ( ,S'~ /ij cqbTs ~ rj 
 
 Ian A h = li _r~^f^ '^' <^osjS~Jl V I 
 
 V Voo. I .V _: A, cos W^'ci) )■ 
 
 Inn 
 
 
 (13) 
 
 J '"'''>'' nlmli'iif cfh.ilf t 
 
 "•■ne. ,/•//.,.,,/,,_ • ""■''""f"f '""■!' -^I'h In t,nns of fke co- 
 't iiiiist be reitu-Jr J tl t 
 
 : ;;; o ^^f -> -^ -'"^1. denJ,; V7:1m "f ^' '-'^ ^-« than 
 
 '"'•"• ^""l".-V(.nt,.ach-)ro=. ,h.it i, M t^' "'ill always ov- 
 
 ^7''^- -^'--..Mho.:.o; f :^"^•^^r''^'■^"^^''■^'-- 
 ■+"^ — ^ /6's» than \m' 
 
 Wlliillcc 
 
 J'Jiei'Hbr. the. formnl.TlOi ,,.7 , 
 
 '■" '—■" -■C;"J:lc:'Sr*' 
 
 ■'"'^ thethinl bvMn',.eobt ntin "" "' "^« «-°«*^ ^3" sine * 
 »:« "I^''^'" ''^'''^^' '''l"^t,ng the first members. 
 
 
 ■ V 
 
-178- 
 
 sin A _ sin B sin (J 
 sin « ni.n b ~ sjn c 
 
 lU) 
 
 n I 
 
 whence 
 
 or 
 
 sin a sin yy = sin /' sin ^4 n 
 
 in h sin C — sin «• sin /i ' 
 
 sin <• sin A — sia a sin ^' ) 
 
 sin «: sin /': sin r = sin ^4: sin H: sin (7 
 
 (151. 
 116) 
 
 :,i 
 
 that is; /Ae .sZ/ics nf Hir gii/rs arf /irn/Kirtiniin/ /<< tfr .fiin's of the 
 oppositi! mu/lrs: 
 
 T,, ^ . , siiwl . nu^ ^ 
 
 Ihen'tore sin a — sin f> -. — - - sin r 
 
 sin /> sin (' 
 
 . , sin B sin Zy 
 
 sin /' :■- SlUd — - =: sin '■ 
 
 8IU .4 sin C 
 
 sin C . , sin C 
 
 sin '" — sin " ---■ — sin '< 
 
 sm A sin « J 
 
 (17) 
 
 . , sin (( 
 Oonvi'i'sdv sin A — sin /) — sin v 
 
 sin '' sin < 
 
 sin ./ 1 
 
 ,. . . sin A sin A 
 
 ><in /y = sm /I = sin (' 
 
 sin t/ 
 
 in (• }• 
 
 (18) 
 
 • -, , sin , sill I- 
 
 sm (' — ■ sill .1 _ ,siu u 
 
 sin (( sin '/ 
 
 Multiplying tho tliinl ut.l. I.y ,;os A/and s„l,stit.iting thu second 
 nmnlvr ot iho ivsult in the- tirst of 1 , wo ohiaiu fitter transposing 
 
 cos (( cos' b 
 
179 — 
 
 cos a — COH -~v cos ' 6 •= coh A siu Asin c -\- coa C sin a sin <; cos li 
 But cos (( — cos '/ cos b = cos (M I — cos'-' ht = cos a sin' /j 
 Substitutin<,' sunl diviiling liy sin l>, wc obtuin 
 
 cos « sin /i — cos .1 sin <; -f- cos C sin n cos /' 
 
 Dividing oach tflims by sin a, ami substitutinj? tor 
 
 sin (• . , sin '',.., ,. , . 
 
 -7 Its einml . 16i, it bccom''s utter ri'diiction 
 
 sin a -^m A 
 
 cot a sin /' = cot A sin 6'-|- cos C cos h 
 Since thb last formula is i^cucral, wo may iutcrehanije the letters 
 ", li, c, and, thfirefore, A, B, t\ ami wi- obtain the following jom- 
 plote system; 
 
 cot '/ sin li = cot A sin C+ cos Ccos /» (19) 
 
 cot <( sill (• = cot .1 sin li-^ cos H cos •• i20) 
 
 cot li sin (I — cot li sin C'+ t^os C cos a cJl) 
 
 cot /> sin c = cot 7j sin A -f- cos ^ cos c i22) 
 
 cat c pin n = cot C sin 7/+ cos Ji cos « (33( 
 
 cot c sii /; = cot C S'.n .1 -|- cos A cos i (24) 
 And by the polar triangle, we obtiiu also the toUowing system: 
 
 cot A sin B = cot a siu c — cos c cos B i26i 
 
 cot .4 sin '' = cot rt siu /' — cos b cos (' (»J(J , 
 
 cot Ji sin .-1 --- cot b siu r — cos c cos .1 (27i 
 
 cot B sin 6' = cot b sin c — cos « cos '"' i'j8' 
 
 cot C siu .4 = cot c sin 6 — cos b cos A i29' 
 
 cot Csin Ji ~ cot r sin a — cos a cos />' (301 
 Niiftit'r's Aiitdoijit's. 
 
 **■ *l 
 
 ting the second 
 :er transposinu; 
 
 Dividing and multiplying tlu> lirst equation o\: 6i by the second 
 give iiwpectively 
 
 tan hA 
 tan yi 
 
 sin (^■— /;. 
 sin (s — n\ 
 
[t^. 
 
 or 
 
 ami 
 
 or 
 
 — ISO — 
 
 tan ^A : tun ^li . : .sin it-- /;) : win i.s " [/'J 
 sin I.s -- ('I 
 
 tan A/1 tan ^i - 
 
 .sin i 
 
 1 : tan ^/l tan ^B : : sin is — ci; sin s [_v] 
 
 By composition and division [f] and [g] buconie, when writtt^u 
 in the form of an equation. 
 
 tan ^A + tan i.fl _ sin i« — ii + Bin i» — <o 
 
 tan ^A — tan ^B ~ ,sin m — 6i — ain i« — en ^ '-' 
 
 1 — tan j^A tan ^B _ sin » — sin i« — ci 
 1 -(- tan j^A tan iB ~ sin .•< + sin [« -— /■( 
 
 ['•J 
 
 By |339i PI. Trig., the second membe;.s of [A] and [<] become 
 respectively 
 
 sin i,s — ^'i + sin {s — a\ _ tan i[(« — bi -f- i.v — a)] 
 •in (s — 6. — sin 14! — a ~ tan ^[(^ — b) — is — a)] 
 
 _ tan |<; 
 tan ^i(i — h) 
 
 sin* — sin 16' — ci _ tan ^[<( — i« — c)j 
 
 sin ,■» 4- sin i* — ci ~ taa ^[$ -f m - «)] 
 
 _ tAU Ac 
 ' taniirt + /;i 
 
 And by (31v'l and i318i PI. Trif;., the first uiombere of [h] and[j] 
 become 
 
 tanjril + ten JZi _ sic kA -\- , \ 
 
 tan j^A — tan ^Ji ~ sin \iA — B) 
 
sin i.s '' 
 
 [^J 
 
 ci; sin s [.v] 
 come, whou wfittt-u 
 
 — a) 
 
 — til 
 
 ['•] 
 
 [//] and [<] beconic 
 
 hi -f- I.V — a]] 
 
 — 181- 
 
 1 - tiiu Ai4 tau Uy __ U08 ^{A + /^i 
 1 + tan i I tun AZ7 ""cos 47^ ITzyi 
 
 Subtititutiug those values in f A] ami f«'] we obtain 
 sin ^i/t -f- iii _ tau Af 
 
 And 
 
 .sin hiA 
 
 C08 ^1.4 + B) 
 
 tan ^(« — 6i 
 tau if 
 
 cos ^ A — /^i tau Ai« + <>i 
 
 or sin ^\A -\- lit : »iu \ < A — Ii\ ; : tau ^c : tiiu jia — t) (31) 
 and cos hA -4- /i) : cos hA — li : ■ tau ^t" : tan ^\a -\- b) (32i 
 which are known ns the tivHt and second Analogies of Napier. 
 
 By the polar triauglo tlu' two preceding formula; are transformed 
 into 
 
 sin h\a ~\- III _ eot ^C 
 sin A;rt — // 
 
 cos }>\(t 4- ^1 
 
 tan A..4 — B\ 
 cot ^0 
 
 or 
 
 cos ^,a —b) tan i /I -+" h, 
 An ^ \ ■ In : sin ^[a — //i ; : cot hC : tan i^iA 
 
 B\ (33, 
 
 cos ^i« -} h\ : CC3 ^[a — !•) ; ; cot iC : tan hA + li) i34) 
 which ixw known as the third and fourth Analogies of Napier. 
 
 Particular Case of liiijht- Angled Uphericjl Trianyle. 
 
 Inibers of [h] and[(] 
 
 The formula' which have been established id the foregoing pages, 
 assume, when applied to the liight- Angled Spherical Tiiangle, sim- 
 pler forms, some of the terms vanishing in consequence of contain- 
 ing the cojiiie 01 cotiiugtnt oJ the iingle. 
 
 Thus if A be a right angle we have cos ^4 = 0, cot j1 = U, and 
 sin .1 — 1. In this cuse, the first and third of (ir)) become 
 
' \ 
 
 I* 
 
 -laa— 
 
 .J 
 
 . ' 'V 
 
 [•29) and i'27l become 
 
 yheucf 
 
 sin /' = «iii " "i" ^* 
 sin '• = ^iu a sin C 
 
 cot 6' = crt /) sin /> 
 col B - cot '' «iu '• 
 
 sin /' — tun c cot C 
 sin (• — Inn '' cot li 
 
 Tho first of a Ihf comes 
 
 cos (( = COS /' cos (• 
 
 |3r»t 
 |3S) 
 
 (37) 
 (38) 
 
 i39i 
 
 The first of i9i nfti'v tiic Ivansi-ositiou of the last term ami th- 
 division by sin li sin (', brconius 
 
 (•us '( - cot li cot < ' 
 
 40i 
 
 Tho second and third of ("Jl In-co Uf 
 COS />' = cos h 
 
 cos />' = cos li sin (' . 
 cos (' = cos '■ sin />' 
 
 (•20 being ilivided by cos c, .md 19i by cos b, become 
 
 cos li = cot a tan r 
 cos r -- col '( tan /' 
 
 (41> 
 
 1 4-2' 
 
 (43 
 
 i44' 
 
 If wo consider the ten equation.s above established, we sec 
 1st, that the first members ad contain <nther a sine or a cosine: ;inil 
 the second members contain i'itiier ;i iiroiiuct of a tant?entintou 
 coUingeut, or a product whose factors are sines or cosines; 
 2ndly, that in the lirst members, I lie two sides iucludiut,' tlie right 
 angle, are always expressed by tlieir sines, and the other three p irts 
 are always expressed by their co«iues: and m the second members 
 the two'aidfis of the right angles are always expressed by their 
 cosines or tangents, antl the other three parts by their sines or 
 
 cotangt 
 meinbei 
 second 
 the colli 
 selves, 
 first luei 
 'he secoi 
 wiiere n 
 
 I'he rig 
 
 it remains 
 
 'lUL-ut'.y IV, 
 
 liie tive pm 
 
 (•arts wiiic 
 
 •"■P cmlud u 
 
 it now w 
 
 ''.i^Uie, obsi 
 
 We SUB tliiit 
 
 the lun^fUI 
 
 qua] tollic 
 
 are aJuays , 
 
 portimt folJt 
 
 inventor, an 
 
 •^'ii' in tli(. p 
 
 
-Co«d ,„e,nl.e... winch eau b 1 ' '^'T" ''' '''"'^""'-"' '^^^ 
 ^te co.n,.lemenu ul tin- ...t, , !; ' r"^"^' ^*^"^' ^^ ^^""pJy taking 
 -Ivc, «--i-iiobtaa,4.: , ;/■ , ;"^^-^ "' tlu. pari, them 
 
 »i'««econa. u.s„,ay 1., s.-.a ,''*' ^'^ "'''"»'''"'"•'"'• two conine. in 
 
 sin </ = siu ,, ^, ^^, 
 ain c ^ Miu „ ^j^ ^, 
 sin <; = u»u , cot I ■ 
 •■*iu '• - liiu ,. out n 
 cos u ^ co.s ^ eo« f 
 
 cos «= cot /y cute 
 eo«iy=:coa^«ju C 
 co« c; ~ uyj, ,. y-jj ^^, 
 
 cosi/ = eot„tau.- 
 Ws V ^ out ,^ j.^y ^^ 
 
 '•'" ''' = COM ,, 008 Z^' 
 
 »'"'• = tosacoa t" 
 ■^'" '> - tan c- tun < " 
 ■■*iii '■ =: tan /, tun h 
 sin u = eus ,, ^.^J|J ^ 
 
 siu « = tun i/ tan V 
 
 sin/r'_ cos ^ cos c" 
 
 sill C -^ cos C COB i^' 
 
 sm/i = tan « tan t- 
 sin ( ■= tan « tan 
 
 'i'iie rigiu aiiyi^, |„.,y„ 
 " '••^■""ms iu the t..anf;;e tivrj'u.s ""''""''"'' ""' "' '^""^idemtion 
 
 '"•"« -"'-' lu. iinmed.at , """/'^"-"'^-•""d '^^n the two 
 ^"•« called u^ijuceuc ,urt. 1 ' i " '''" "«^^ ""^ ^"^ "' it 
 
 ,. ^' "-V we oonsuLu ; ::':"• ':- '"■•• -'^-^ <^PPo.ite,.ru: 
 "-^"'■c. obser ,ng that „ ^ ^ .. " ? '"^'"^'"^'-^ vvith the last 
 
 '^'" ^-^-' ot two uthe. Z V:; '^ 7'"^^ '^ ^'^ P-duct oi' 
 H"'*' tothe ,.,.u.iuc., ui ,1.,,.. :■ ^ ^;^''^' ■ ^^'^' '^^"-^ 01- any part ise- 
 -;'-.^ '^..... :; ■ ™:: ^^7 -"•■■• P-. the. lastparta 
 
 "'vomo,., and which is snml^ '"'"' "■'^"' '^''^ "'^'"'^ of tho 
 
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 Sdences 
 
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 23 WIST MAIN STREET 
 
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 (716) 172-4503 
 
 

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 N(ii>iir\<. liiih'. 
 
 The nine of any paif '» d riglil-ainjli'il S/ihcrical Triiuvj' ■ in »•- 
 qual to the product i\f the tuinieiit^ of (he tiro ailjarenl jiartii, irnf 
 the roitmeK of the two npjio>>if'- j.iirtn , 
 
 Olilif/tip Aniilf(.l Trliiidile (tiiil Pfi'iicndi'-u'.iir. 
 
 ■!-■ 
 
 '^■.■*;;s 
 
 ' Vl 
 
 If a perpeufliciilar bo drawn from any augle in an Oblique An- 
 gled .Sjihcrioal Triiingle, sonii' other uaeiul formula' may be obtained 
 Thus lot Cf be pei])endicular on AB, in the annexed figure. De- 
 note tlie segment --1 1' by v 
 and the angle ^46'/* by I', 
 then the otlittr segment 
 HP- Ali—AP=e ~ r, 
 the angle li<:P = t'— V. 
 liy (43 wi! have in the right- 
 angled 'rrian,le AVI' 
 
 cos A = eot h tnn /■ 
 fl'lience, .-jinco tangent and 
 coting^nt are mutually re 
 eipvooal, 
 
 tan /•= tan /- (xk A \ 
 Similarly tan \c-- r ~ tiin c eos /; \ i45, 
 
 that is: the tamjeiit of either .-v/wr'/d' nf th la^e is rqiinJ to the pn- 
 duct.of the tangeni of the adinreiit sid e int- the cosine of the adia- 
 cent atif/le. 
 
 By (40i, 
 
 cot '/ = cot I! cut I ' 
 
 jfrhence 
 
 cot V = 
 
 cot (/ 
 
 eot5 
 
 whe 
 
—185^ 
 
 'icrical Triaivj' ■ in c- 
 oifjfiiriiij'drfs, ai'iif 
 
 '(■ndi'-uliir. 
 
 it! in an Oblique Au- 
 
 iiulii' may be obtained 
 
 annexed figure. De- 
 
 i45. 
 e is cqvnl to the i,r(- 
 ' com)!' nf tie mlid- 
 
 Taking the reciprocal, 
 
 tan r=:^°i£ 
 cot « 
 
 Similarly tan , C— V, ~ ^'^^ -^ 
 cos i 
 
 I46i 
 
 By (39/, ^ ^^ tt^acent side. 
 
 cos ft = cos w cos CP 
 
 cosa = co8(c-.,;eo,CP 
 
 '=°' « "~ WT^:^ ,47, 
 
 ^^'''^ ^^' the cosines of th^ .. 
 
 ''' . «i°^ = cot^ta,6>/> 
 whence ^^^J ^ cot^ ^ ^^^ ^ 
 
 ^•■^'^^""^t-:^ 
 
 that is; ^^^ 5,- 
 
 J^y(4i), 
 
 whence 
 
 cos ^=„-„ j^^^^^^ 
 
 ^?^ ^ sin r 
 
 (49, 
 
 *hatis; the sines of tho 
 
 f> II 
 
Ill ■■ ; •,.■ '.K 
 
 —186— 
 
 M :»^ 
 
 lij i44i, cos V= cot b tiin CP 
 
 cos (C— 7) = cot c tan CP 
 
 cos V cot /> 
 
 whence 
 
 cot a 
 
 tun a 
 tan ^ 
 
 |50i 
 
 cos (C— F) 
 
 that is, the cosines oftheiiurtial umjles made Inj the perpcndiciihr 
 are directly propwtiona Ito the cotc.iujents, or inversely proporilGu- 
 al to the adjacent sidns. 
 
 By following a similar piocos;;, wo dcdiicc, comparing tlit^ two 
 partial triangles, 
 
 tan ?• _ tan V 
 
 tan (c — n " tan (C — V) 
 
 that is, the tangents of the tegmenta of the base are proportional to 
 th» tangents of the partial angles. 
 
 CHAPTER II. 
 
 ■U- ,. 
 
 Solution of Spherical Triangles. 
 
 If any three of vhe partaof a Spherical Triangle he giren, the 
 other may he determined, and it is not necessary that one of the 
 given parts he a side, as in Plane Trigonometry. 
 
 Riiiht-Angled Spherical Triangle, 
 
 Vrhen the lule of Napier is made use of in solving a right-angled 
 Spherical Triangle, all the cases that may occur in practice, are 
 found included in the following one where the right angle is not 
 taken into consideration, as heirg always known: 
 
 Being given any tieo parts, to determine the three other parts 
 For the two given parts are either adjacent to each other, or sep. 
 amted by another part. And in every case, they must he either ad- 
 jacent oi' opposite to, at least, one cf the unknown parts v/hich be- 
 ing determined may serve to determine the othoi-s. 
 
 It must be ol)served that when the 'Quantity sought is determine 
 
.-as7- 
 
 •' "I I 
 1 
 
 (50) 
 
 c perpcndiculir 
 rsely proportton- 
 
 iipatm: 
 
 ft the* two 
 
 re proportional to 
 
 b* giyen, tlio 
 ry that one of the 
 
 le. 
 
 nng a right-angled 
 
 ,T in practice, are 
 
 right angle is not 
 
 i: 
 
 three other parts 
 
 each other, or sep. 
 
 must be either ad- 
 wn parts y.'hlch be- 
 
 ire. 
 
 sought is det«rmine 
 
 cd by itj sinu as is always the case in the use of Xapier's rule, it 
 will be ambiguc-'s wiiethor the quaulity is loss or greater than 90°, 
 since th i sines are positive in both the first and second quadrants. 
 The ambiguity nuy liowever generally be removed by the following 
 priaciples. 
 
 I. Jii II Jiii/ht-Anijlid Spherical Triangle, an obliqve amjh and 
 its opim&ite side are always of the same species, that is, are either 
 hoih lesser or botu i/reater than 90°. 
 
 II. When the two sides including the right angle are of the same 
 species, the lujpvtennse is less than l'()°; and irhen they a re of dif- 
 ferent species, the hypotenuse is greater tlian 90°. 
 
 The first of these principles follows from (37), where, since siu h 
 jS positive, tan c and cot U mu.st have the same sign, and, there- 
 fore, c and C must be in the same quadrant. 
 
 The second i)rinciple follows from (39), where, if h and c are in 
 the same quadrant, their cosines have the same sign, and therefore 
 cos a is positive, that is, a is less than 90°: if on the contrary, b and 
 c are in different quadrants, their cofines have different signs, and 
 therefore cos a h negative: that is, a is greater than 90°. 
 
 Morei>ver, another form may be given to the rule of Napier, so 
 that it leaves no ambiguity. For this puipose, instead of taking in 
 the group on page 183, the sines for cosines in the first members o^ 
 the equations, tlie tangents and the cosines for cotangents and sines 
 in tlie second members, take the cosines for sines in the first mem- 
 ''"rr-, and the cotangents and sines for tangents and cosines in the 
 aecoufl uiembers. Thus the five part-i, instead of being, the two 
 sides includiuu; the right autjle and the complements of the other 
 three parts, are, conversely, the complements of the two sides inclu- 
 ding the r'ght angle, and the three other parts in their primitive 
 ■ondition. In this case the rule may be enunciated as follows: 
 
 Napicfs rule modifii'd. 
 
 The cosine of any jiort in a Right-Anghd Spherical Triangle is 
 equal to the praditrt (f the cotiiuyciifs of the adjacent partsorofthe 
 ^•ines if till' ni'i'iisi/t' /"irfs. 
 
 The (juautity .ought being, by the last rulo, determmd from its 
 
-lljS^ 
 
 ' !'■ -1 
 
 1 
 
 
 'h 
 
 ■ I 'M 
 
 
 cosine there id no ambiguity whetlier it belong to the ilrat or tliesec- 
 uiul qunnilriint. • 
 
 'J'ho chief ailvimtaije of Kiipiers rule is the great lielp it gives to 
 memory, lini in empi Dying it, sonje inconvenience arista in so far 
 as a reijiiired itsit may not always be determined independently of 
 tlie other unknown parts, which then must be lii-st determined, 
 by employing independently of l^Iapier's rule the ten equations pre- 
 viously established wecaneasily determine, in eveiy case, any one 
 uf the three unknown parts independently of th) two others. 
 
 Then the solution of a Kight-Augled Spherical Tiiangle presents 
 the six fallowing cases: 
 
 I. Then the hypotenuse and one oblique angle tae giVen. 
 
 II. When tiie hypotenuse ami oi.e side are given. 
 
 III. When one oblique angle and iti opposite side are given. 
 
 IV. When one oblique angle and its adjacent sid« are given. 
 
 V. When the two sides are given. 
 
 VI. When the two obliqu.^ anglen are given. 
 
 W^e shall proceed to show how each of these cases may be solved 
 by liapior's rule, or Napier's rule modified, and by independent 
 i'ormulie.. 
 
 Let the student be very particular to take tlie complement of the 
 hypotenuse and of the two oblique angles, instead of these parts 
 themselves, when employing Napier's rule; or the complement of 
 tliR two sides including the right-angle, when using Xapier'M rule 
 modified. 
 
 An accent annexed to the symbol of any part denotes, sis previous- 
 ly stated, the Ci'mplenieut of that part. 
 
 '':■■ I 
 
—lio- 
 
 loliMtovthesec- 
 
 • 
 
 , help it givea to 
 c arisfcs in so i'av 
 ndependently of 
 xnt determined, 
 len eiiuations pre- 
 eiy case, any one 
 wo others. 
 iTiinngle present* 
 
 l«i av«i giVen. 
 
 'en. 
 
 ide are given. 
 
 id« are given. 
 
 ases may be solved 
 H l)y independent 
 
 Icomplementofthe 
 ^.otid of these parts 
 the coinpleinent of 
 
 |ising Xiipicr's rule 
 
 fjnotes, as previous- 
 
 Case I. 
 
 Given the hypothenuse «,and tlie anjjle B, to find h, c, C. 
 
 1. Solution hy Napier's rule. 
 
 sin /> = co8«' COS />"■ (f)]) 
 
 »iu « - Un fj tan Ji' if);') 
 
 sinC = tan«' tiin A (fl3) 
 
 2. Solution, hi) Napier's rule utodijietJ. 
 
 cos // = sin a sin li 
 cos c' = cot // cot Ji 
 cos C - cot a col 1/ 
 
 (54 1 
 |56) 
 
 3. Solution by indeneinkui turmulo; 
 By (3fi) sin /; = sin a sin B 
 
 iiy i43i, tan r .= tan o cos ii 
 
 By (40 i, cot C = cos a tan B 
 
 (57) 
 (58) 
 (59 1 
 
pi' ' ' ■* 
 
 I 
 
 —190— 
 
 Caae 2. 
 
 IM 
 
 (liven the hypoteniusp /. and one sidn <,'. 
 
 1. , Solutioti hy Noj)ier^a rule. 
 
 sin W = tan li ton <■ 
 sin 6'' = C08 c C08 2^' 
 sin b = cos a' coH 6" 
 
 2. Solution hy Najiier's rule iiuxlijii'd. 
 
 coH /i = cot a cot c' 
 coa C= sin r' MJn B 
 cos // = sin a sin 6' 
 
 1 00 
 i61l 
 1 6-2 1 
 
 |G3| 
 611 
 1 65) 
 
 3. 
 
 SolufioH 
 
 % 
 
 liulepcntlcu t form, uln: 
 
 
 By A3) 
 
 
 „ tivu r. 
 
 cos /J = 
 
 tiin a 
 
 
 By (36i 
 
 
 sin C; = . 
 
 Hin « 
 
 
 By (391 
 
 
 cos 11 
 
 cos /< = - — 
 
 |6G) 
 |67) 
 i68) 
 
 cos '; 
 
 •f 1 
 
 (!ase 3. 
 
 Given one oblifiuc unglo B and its opposite side h, 
 
 1. Solution hy JVapier'a rule 
 
 sin (• = tan L tan B 
 sin a' :-• cos h cos c 
 aiia C = tan « ' tan b 
 
 (69) 
 
 (70) 
 (71) 
 
iflO 
 i6l) 
 i62) 
 
 —191- 
 
 2. Solution b>, Napier's nile. modi fied. 
 
 cor, ,•' = cot b' cot B 
 co.s a = 8Jn b' sin c' 
 cos C = cot rt cot b' 
 
 3. Solntlou hji 'iif/e/xndent /omulcc. 
 Rv (3«i -:- tan b 
 
 sin « = 
 
 tun li 
 
 (72) 
 (73) 
 174) 
 
 (75) 
 
 
 (68) 
 (65) 
 
 % (35) 
 By (41) 
 
 sin a = 
 
 sin C = 
 
 ain b 
 sin J? 
 
 cos ^ 
 cos 6 
 
 (76) 
 
 IM' 
 
 |6G) 
 (67) 
 
 (68) 
 
 iide h. 
 
 «an,e i„ bo^h, but of which ik^^emZ^^,, ZlT "'^^ *'^ 
 angle of tho one are tho supplements of fhn , remaini.ig 
 
 remaining angle of the otho^ C flit I ""'^° '''^'' ""'^ ^^- 
 ca«e, the given parts a,, on id Id t^f ^"' "'^ the preceding 
 iem admits of two solutions .mdfhf '"^ ^^'^ ^rig]e, t).e prob- 
 
 ItnuistnotbeinferrlrLttet^Zet^u 
 le.« th:in 90° in on.3 trianr,i„ , required parts are 
 
 l-MUe, given on U« pa!; /jr'' "'*' •°'''°" ''^ ""^ »' "■« 
 
 
 16O) 
 
 (70) 
 (Tl) 
 
 Case 4. 
 
 ^--T°^^^<i--.^IeZ.. and its adj-acont side 
 i-, ^o/tt<jo„ f.y ^-apier's rule. 
 
I\ 
 
 
 ti:' ■.; 
 
 Id' 
 
 I- f . < 
 
 — 192— 
 
 win C = (JOS c cos /?' 
 
 Hin «' = fan/i tim T' 
 sin ^y = cos ii' cohZ?' 
 
 2. Sol lit, 'on bij Niipier'a rnk modhivil, 
 
 cos C = sin «' nin B 
 cos a = cot 5 cot O 
 cos ft = ain n sin 7? 
 
 3. Solution hy independent formula: 
 
 By (42l cos C = cos c sin ^fl 
 
 15y (43| cot a = cot c cos 5 - 
 
 By 1 38) tan b = ain f^ tan B 
 
 Case 5. 
 
 Uivon the two sirles b, r. 
 
 1, Solution by Napier's rule.. 
 
 sin a' ~ cos 6 cos c 
 sin 5' = tan a' tan /• 
 sin (/ = tan a' tan ft 
 
 ?, Solution by Nnpivr's'rule modified. 
 
 cos a = sin ft' sin «' 
 cos JS = cot a cot I'' 
 COB C = cot a cot ft' 
 3. Solution by independent fcrimlw. 
 By (39) CCS a = cos ft cos c 
 
 By (38) cot J? = cot ft sin ,: 
 
 By ,87) cot (7= cot r sin ft 
 
 (7Si 
 
 (83) 
 
 (841 
 (85) 
 (86) 
 
 Giv( 
 
 ^o/m// 
 
 3. Soluti 
 By (4( 
 
 By (41 
 By (42 
 
 (87) 
 
 N. b. 1. In 
 partii which ex 
 
 (88) 
 
 tangenta and 
 
 m\ 
 
 esUtbllghed on 
 i In I 
 
 
 b« particular t 
 
 
 1. Gi^ 
 
 (90) 
 
 
 (91) 
 
 ■ 
 
 (92i 
 
 2. Giv 
 
 (93) 
 
 
 (94) 
 
 
 (96) 
 
 
(78) 
 •79* 
 
 m) 
 
 m 
 
 (88) 
 
 (84) 
 (85) 
 (86) 
 
 (87) 
 (881 
 (89\ 
 
 (90) 
 i91) 
 
 (92: 
 
 (93) 
 
 (94) 
 (951 
 
 S^ 
 
 —193— 
 
 CuMe 6. 
 
 Given the two oblique angles li, c 
 1. Holution h,j Napier't rule. 
 
 sin ii = tau li' tan C" 
 in h = C08 a' co« li' 
 sin c ~ cos li coa C 
 2. ^fl/«/;o« ty iVtf^/sr'-* r../e vwdifitd, 
 
 cos rt = cot li cot C 
 COS 6 = sin a sin /i 
 cos f = sin a sin 6' 
 
 3. Solution by independent formulos 
 
 By ,40) . 
 By (41) 
 
 % (42) 
 
 cos rt = cot /? Cvt C 
 
 cos U 
 sin C 
 cos 
 
 cos A = 
 
 COS c = 
 
 sin iB 
 
 Exercises. 
 
 06i 
 
 (97) 
 
 i98j 
 
 (99) 
 (100, 
 (101) 
 
 (102) 
 (103) 
 
 (104) 
 
 
 1. 
 
 2. 
 
 Given a = 91° 4-2' 
 B = 95° 6' 
 
 Given « = 140° 
 
 c = 20° 
 
 Anstcer b = 96° 22' 30" 
 c = 71° 32' 14" 
 C=71°36'47'v 
 
 Answer B = i2° 8' 48" 
 C= 115° 42' 24" 
 ^~ 144' 36" 28° 
 
 :i 
 

 
 • 
 
 
 
 » 
 
 
 
 f 
 
 1 
 
 --l'J4- 
 
 
 1 
 
 1 
 
 8. 
 
 (Jivpii Ji = :J(i« Angnrr ,• - 42"' 1!)' 17" 
 /- - -.'(i 4' n - 46' 22' :i2" 
 
 C' = (14° 14' 2(5 ' 
 
 
 'i 
 
 
 01' ,• = i;57" 4()' 4.1" 
 
 
 \ 
 
 « = i;u° -M' 8" 
 
 
 4. 
 
 S. 
 
 6. 
 
 Givoii n - 44° :>(»' 
 
 Given /;= lie 
 ,• :.- Hi 
 
 (iivfu Ji = (5!)^ 20' 
 
 c. -• iir>° 4r)' :i4" 
 
 Axx'irr r'=Gr.° 4i)' 63" 
 (« =: G3^ 10' 4" 
 h:=W° .VJ' 11" 
 
 AiiHirir n = 111° :,.-)' 20' 
 
 /; - i)i' 'vy -.'4' 
 
 (;- 17° 41' 4(1' 
 
 Aiisirn- 'I — 7li' .1(»' ;»7" 
 // ^- (55"' -JS' r,H" 
 r - U:y° 47 4<r' 
 
 (Jini,lninfiil Triunijit'. 
 
 A (itiiadrajitiil Triaii^'le Ih a aphcrieal 'Irinunit! liaving oni- of lla 
 siiio a quadrant. '1 ho polar trianglo of the HifjIil-AnylcHl Triangltsiw 
 therefore ahvay.s a (luudrantal triangle, Such tri ingles are generally 
 avoided in practice. AVhen unavoidable, they iimy he iciidily solv- 
 ed liy means of the jiolar triangle. 
 
 Oblli/iir Aiii/lt'd Sji/icrifil TriKmilr. 
 
 The solution of an (Jl)liquo -Angled Splierical Triangle present 
 the six following cawss' 
 
 T. When two siJos and the angle opposite to one of them are given. 
 II. When two angles and the side opposite to one of them are 
 ''iveu. 
 
= 42 l'.» I' " 
 = AH" 22' :»2" 
 : (14° 14' W 
 
 - ViV 40' 4."^ " 
 
 : i;n° :^7' f< " 
 r. iir)° 4r.' :i4 ' 
 
 = Gr)° 4!t' B3" 
 = G3^ 10' 4 " 
 = W .V.)' 11" 
 
 = iir V)' 'JO' 
 r- \r,' -vy -M' 
 
 - 17° 41' 4(1 ' 
 
 =.. 7(r :i<i' :»7" 
 ^ (jj'^ -JH' r.s" 
 
 - :.r)° 47 4<;" 
 
 .iL!,lf liming one of ita 
 
 Iglil- Aiigit'il Triangli) w 
 
 tiiinyles live i^cui^riiily 
 
 iiiiiv be icmlily f'olv- 
 
 -1 sa- 
 
 in. Wlieii two Hides uudthe iucludud HUgifl «i« givou. 
 
 IV, Whon two angles aud the included side are giv«n. 
 
 V, When the three sideH »re given, 
 
 VI, When the three angles are given. 
 
 Case 1. 
 Oiv«n two aides, a, b, and the angle A oppoute one of them 
 
 By (181 8in£- aiu i4 
 
 •in b 
 8>n a 
 
 Then by Napiev's Analoj,', (31 1 
 tuu 
 
 , sin ^ ^ ^- Ii\ 
 
 (lOol 
 
 (106) 
 
 Iriciil Triangli' im-bent 
 
 I one of them are given, 
 to ont^ of them arc 
 
 /; could be determined also by any one of the thi«e other Ana- 
 logies. ,. 
 
 sin C— tin A 
 
 81U c 
 
 sin a 
 
 (107) 
 
 Therfi are generally, in the preceding case, two solutions each of 
 which will satisfy the conditions, when the required part is detor- 
 ntined by its sine. 
 
 t 1 
 
 i' 'I 
 

 ■ 196- 
 
 TIio aiiihiguity m\y be roino ml hy tlio i'ollowiujj itviiioipluji; 
 1. Tlio greator shlc. inusl lio oniosile tlic gieateinnglo, ami C(»n- 
 
 vo"sjly. 
 2 When the buiii of the given sitlea is* losfl than 180^, the an- 
 gle cppjnito to thi li« ditl) U iiC'it!. 
 
 3. When the sum of the two given sidoa exceeds 1 80°, the an- 
 gle opposite to the greater side is obtuse. 
 
 4. WheJ the sum of those sides is 180'*, the i>um of the oppo- 
 site angles is the same. . 
 
 5 When the given side not opposite to the given angle diflers 
 from 90° more than the g'.ven side opposite to it, the angle opp*. 
 site to the lust side must be in the same quadrant as that side. 
 
 When thise principles, which are based on Spherica'. Geomeliy, 
 fail in determining the species of tiie angle, it lemaius doubtful. 
 When C only is ruquiied to bo known, it can be determined from 
 
 (45) and |47', which giv,) . _ 
 
 Ian V — tan l> cos A 
 
 cos a 
 
 cos (' — r\ — — cos I 
 
 cos h 
 
 1108/ 
 llOi)* 
 
 3. 
 
 c will be the sum or difference of v and (c — v\ acconling m A 
 and B should be of the name or of different species. If li is doubt- 
 ful, c will he also doubtftil, and, therefore either the sum or differ- 
 ence may be taken. » 
 
 When C only is refjuired to be known, it may determined from 
 (40) ami iflOi whi;h give 
 
 tan F = 
 
 cot A 
 
 COi 
 
 aioi 
 
.^197- 
 
 j»lo, and con- 
 ,«0'', the an- 
 180°, tho an- 
 il of iVie oi>po- 
 
 1 angle differs 
 tie augli opi)«- 
 i as that side. 
 
 ica'. Goonicliy, 
 iu8 doubtful, 
 iteriniued from 
 
 (I08i 
 
 laccoiiUng ai A 
 If li is doubt- 
 lie sum or differ- 
 
 L-termined from 
 
 aioi 
 
 r< I 'I *"" ^ ir 
 
 COS iC — I I = — COM I 
 
 tan a 
 
 (HI) 
 
 ('■ is tho svini or the difference of V and i^^ — V) according as 
 A and li are of the i^ame species ov not. \\ U is doubtful, C'is al- 
 so doubtful, and either of the two values may ho taken 
 
 Case 2. 
 
 Given the two an'.;h'S A, B, and the side a op, osite to one of them 
 
 1 11 'J, 
 
 r, 1 ^ • I 81" ^ 
 
 By (17i sin ft = sin a —. 
 
 Tlien c and Care found as in 10+) and (lO.'ii 
 
 This case also admits of two solutions, except when tiu^ ambigui- 
 ty is removed by tlie following i)rinciples; 
 1 . When tho sum oi the given angles is hvss than 180°, the side 
 
 opposite to the less is le.ss than !)0°. 
 'i Ifth .■^11111 of those angles exceeds IWP, the side opposite to 
 
 the gi-eater is greatt-r than ll(»°. 
 3. If the suTi of those angles is 180 ', tho sum of the opposite 
 sides is the same.. 
 
 If c or C only bo requiiwl to bo determined, operate with the for- 
 niulie 4") (the first) and '48) for the former, and with (48) and iflO) 
 for the latver, in the same manner as before. Tho arcs v and 
 \r, — V] are of the same species or not according as A and li must 
 he; 0, will be equal to the snm ordifference of « and [c. — v\ aceoiding 
 as A and li should be of the same species; T'and iC — V] are of 
 the same species or not according as a and h should be; and the 
 sum ordifference will be taken for the value of c according us A and 
 B are of the same or of different species. 
 
 I ! 
 
 ) 1 
 
1.. I. 
 
 ri" ^ 
 
 Case 3. 
 
 Given two sides <*, b, and the included angle C. 
 By Napier's Analogiej' |31) and i32) ^{A+Iif and ^(^4 — Ji> 
 are determined. The sum of tliese I'esnlks is the angle opposite 
 the greatc.1 of the given angles; their differcnee in the other angU. 
 Then ^c may be determined by either of the same Analogies. 
 
 I{ e only be required to bo determined, proceed with (45 1 and 
 (47) as before. 
 
 Case 4, 
 
 Given two angles A, B, and thj included side i\ 
 By Napier's Analogies (33) and (34) i(a-|-'') and ^ia — b] are 
 
 determined, whence a and h are easily known. Then ^C is found 
 
 by either (33) or (34). 
 
 If (7 only be required to be determined, proceed with (46i and 
 
 ^50) as before. ^ 
 
 Onse 5. 
 
 Given the three sides, a, b, r. 
 Any one of the angles may be found by one of the formulse \2|, 
 (4), (5), (6i, (7l. 
 
 Ci^s^; 6. 
 
 Given the three Angles A, Jl, O. 
 Any one of the three angles may be found by one of the for- 
 ^ulai (8 , (10), (111, il2l, (131. 
 
 \ 
 
gle C. 
 
 f B) and ^[A — B) 
 i« angle opposite 
 i& is the other angl*. 
 anie Analogies. 
 )ceed with (45 1 and 
 
 aide <■. 
 
 '/) and \\a — b\ are 
 
 Then \C is found 
 
 eed with ^46i and 
 
 jf the formulie v2i, 
 
 bv one of the for- 
 
 7. 
 
 8. 
 
 ■lay— 
 
 Exei'oidus 
 
 f'iven (I ^ 124° tii' 
 
 /y = 3]° 10' 
 
 A-=:-[{\° ^»,5' 
 
 Given o. = i],-)'^ oj^' 
 h = 6()» 29' 
 
 Aiitwer li 
 
 V 
 
 A - 06= 
 
 17' 
 
 ('ivcn ^ - 116° 3G 45" 
 n — 80° 19' 12" 
 /,;.-. 84' -21' 56" 
 
 '•ivcn /I - 61° .37' 53" 
 //^-- 139\';4'34' 
 I' --= 1. 5(^17' ;.'fi" 
 
 ■ '■:'- i2fr' 
 
 (•ivcn A = 104° r/ 
 
 yy^s-j' 18' 
 
 (•ivcll rt ~ 7P ]r,' 
 
 /' = 39^ 10' 
 
 '•• = 40° 3;V 
 
 (Jiven A = 109° 55' 
 
 B=: 116^38' 
 
 C- 120' 43' 
 
 Aiifirrr U ~ 
 or li=i 
 
 Amiipra = 
 
 V = 
 Aimintr a = 
 
 C 
 or It 
 c 
 C 
 A iixwnr A : 
 B 
 
 f ; 
 AII.SICIT a ; 
 /' 
 
 C 
 
 Athiwi'r A 
 
 n 
 
 a 
 
 Aiisivi-r a 
 
 ■ 10' 19' 34" 
 
 ■ 171° 48' -22' 
 l.')5=' 35^ 22' 
 59° 39' 
 172° 43' 
 172° 23' 
 
 : 120° 21' 
 ^ 72" 52' 
 8S° 53' 
 n4°26'o0" 
 82° 33 31" 
 79^ 10' 30" 
 = 42° 37' 18" 
 = 12!)° 41' 5" 
 = S'9°54' 19" 
 = 137" -J-^' 4->" 
 =- 19° 58' 35" 
 = 20° 21' 18" 
 = 62° 3r40" 
 = !8° 57' -29" 
 = 107" 7' 25 ' 
 = 102° ]?■ 
 ~- 86° 41' 
 = 70° 31' 
 --^ l.'^()°36'55' 
 = oO° 25' 34" 
 = 31° 26' 32" 
 = 98° 21' 20" 
 = 109° 50' 10" 
 = 116° )3'7" 
 
 I 
 
 End. 
 
 V/>y. 
 
 vn 
 
 ^>.' 
 
 X