CIS 



MATHEMATICS. TRIGONOMETRY. 



[DEBITED FORMULA. 



Let A O B be the 



angle A; I'"' ' 

 the an^lo B. Then 

 A r : iv angle A 

 + 1?, B O C, and 

 therefore A OC, be- 

 ing measured as in- / 

 dicatedbythe dotted .' 

 circlet. In C take { 

 any point P and 

 draw 1' X, 1' M per- 

 1" inliculan to OA 

 and OB produced, 

 and from M dravr 

 MR and MQ jvr- 

 pemliculars to OA 

 and N P produced. 

 Then 



rs. is. 



Sine (A + B) =. - 



PN PQ-QN 



PQ 

 ~OP" 



Now oS= 



MP 

 OP 



EM 



OM 

 OM 

 OP' 



KM 



SP 



OP 

 MP, RM 

 OP^OM 



OP 



( ' M 



OT 



cos. MPQ = cos POA - - cos A. 



= sin. POM= -sin. B. 



' sin. AOM=sin. A. 



cos. POM =- cos. B. 



.'. sin. (A + B) = sin. A. cos. B + sin. B. cos. A. 

 The student may verify for practice the formulas in 

 the following cases : 

 Sin. (A + B) A 7 90V 180 A + B 7 180 B 7 90 

 Cos.(A-B) A 7 90 Z 180 A + B^90 B790' 

 Cos.(A + B; A7180V270 A+B73COV450 

 B 7 90 Z 180 



21. The principle on which the proof may be considered as 



established generally. 



The above examples will be sufficient to satisfy the 

 reader that the four formulas above given hold good for 

 all angles whatever. It is worth while to observe, how- 

 ever, that, independently of these examples, this follows 

 from the circumstance that the extension given above to 

 the definitions of angular magnitude, and of the trigono- 

 metric ratios, are made in strict accordance with the ex- 

 tension given to the meaning of the negative sign in 

 algebra. Thus, a b primarily signifies that the number 

 & is to be subtracted from the number a ; if therefore b 

 be greater than a, a 6 is impossible, unless we generalise 

 the definition of the negative sign. If we do this so as 

 to render 6 susceptible of meaning for all values of a 

 and 6, then whatever theorem we prove to be true of a 

 6, and its combinations with the restriction, will be 

 equally true of 6, and its combinations without the 

 t ion. 



In like manner, if we prove a trigonometrical formula 

 to hold good for all geometrical angles, those will equally 

 hold good of the angles when defined according to the 

 trigonometrical conception of an angle explained above. 

 The principle which wo have to guide us in all these 

 generalisations, is called " The Principle of the Per- 

 manence of Equivalent Forms," and is that which lies 

 at "the root of all extensions of merely Arithmetical Al- 

 gebra, as explained in the treatise on Logarithms and 

 Scries. (See Chapter VI., p. 512, in this section). 

 The reader who wishes to see a full account of the appli- 

 cation of this principle to Trigonometry, will do well to 

 consult Dr. Peacock's Algebra, vol. ii., p. 144, etc., 2nd 

 edition. 



22. .Relation between the Four Fundamental Formulas. 



It in to be observed that the last throe of the four 

 . formulas given above can be derived from the first of 

 tlicm. 



Thus, sin. (A + B) = sin. A cos. B + sin. B cos. A. 

 For B write B. Now sin. ( B) sin. B. 

 And cos. ( B) = cos. B. 



.'.sin. (AB) = sin. A cos. B sin. B cos. A. 

 Again cos. (A + B) - sin. (90- A- B) 



- sin. (90 -A) cos. B-sin. Boos. (90 -A) 



cos. A cos. B sin. A sin. B. 

 Again 



(Cos. (A-B) - sin. (90- A + B) 



= sin. (90- A) cos. B + cos. (90*- A) 



= cos. A cos. B -f- sin. A sin. B. 



23. Formulas derived from the Fundamental one. 



From these four the following formulas of frequent 

 occurrence can easily be derived : 



Sin. A cos. B -f- sin. B cos. A = sin. (A + B) 

 Sin. A cos. B sin. B cos. A sin. (A B). 

 .' . adding 2 sin. A cos. B = sin. ( A + B) + sin. (A B) ; 

 and subtracting 2 cos. A sin. B = sin. (A + B) sin. 



(A -15). 

 Similarly, 



2 sin. A sin. B = cos. (A-B)-cos. (A -f B).... (12) 

 2cos. A cos. B = cos. (A-B) + cos. (A -f B).... (13) 

 The same formulas are of frequent occurrence in a 

 different form. Evidently 



= o + <t> , 0-0 



~~ ~~ 



0-0 



. sin. = sin. 



cos.?* 



,0_+0 



sin. = sin. g cos % ""^""jjT ' 



+ <A 

 . sin. + sin.0 = 2sin. g c 08 a-*- (14). 



v -)- 

 2 sin. ji~ 







T* (15). 



0+0 



sin. - sin. 0=2 cos. jj 3111 ' 



Similarly, 



(? + (A rt 

 Cos. + cos. = 2 cos. g-^ ' cos- ^ (16). 



Cos. - cos. = 2 sin. -*' cos - 



.... (17). 



24. Formula for the Tangent of the Sum of Two Angles. 

 Again, we can easily derive from the formulas for the 

 sines and cosines of A + B and A B, expressions for 

 the tangents of A + B and A-B. Thus, 



Tn .A - 



Tan. (A - cQg (A 



sin. A cos. B + sin. B cos. A 



~ cos. A cos. B sin. A sin. B 



Divide both numerator of this fraction by cos. A cos. B. 

 sin. A cos. B , sin. B cos. A 



.'.tan. 



sin. A 



COS. A 



sili. r. 



COS. li 



cos. A cos. B cos. B cos. A 

 cos. A cos. B sin. A sin. B 

 cos. A cos. B cos. A cos. B 



tan. A + tan. B 



sin. A sin. li~ 1 tan. A tan. B 



COB. A cog. B 



Similarly, 



tan. A tan. B 

 Tau -< A - B )~l-tan.Atan.D 



.(18). 



. . . . (19). 



25. Expression! in which the Sum of Three Angles occur. 



We can easily derive from the above, expressions for 

 he sines, cosines, <fcc., of the sum of three or more 

 ingles. Thus, 



Sin. (A + B + C) - sin. (A + B) cos. C + cos. (A + B) 

 in. C = (sin. A cos. B + sin. B cos. A) cos. C + (cos. A 

 x>s. B sin. A sin. B), sin. C sin. A cos. B cos. C + 

 in. 15 cos. C cos. A + sin. C cos. A cos. B sin. A 

 tin. B sin. C. 



