ANGLES.] 



MATHEMATICS. TRIGONOMETRY. 



623 



(4). 4tan.-l 



(5). 



-1 _ tan.-* -- = 45 



(6). tan.-im+tan.-' n=s 



(7). If = 



/ m + n \ 



: tan. 1 7 0=tan. i 

 V > 



_ ; then 



sin. (0 + 0) = sin. CO cos. 30 



ON THE USB OF THE SUBSIDIARY ANGLES. 



34 Explanatory. 



In making trigonometrical calculations it is nearly 

 always necessary to conduct them by means of logarithms. 

 For the purpose of preparing a formula for logarithmic 

 calculation, it is often necessary or convenient to ex- 

 press the sum or difference of two or more magnitudes 

 by means of a product : this can generally be performed 

 by introducing the sine, tangent, or some other ratio of 

 an angle chosen for that purpose, which is called a sub- 

 sidiary angle. One or two examples will explain the 

 means employed for this purpose. 



35. Examples. 

 (o) Thus, let x 3 = o 2 + 6 s to find x 



Assume =tan. 

 a 



z'=o 1 (l+tan. I e) 



o 



COB. S 0. 



to that if a and 6 are any two numbers whatever, we are 

 entitled to assume that r- = tan. 0, since tan. may have 

 any value whatever from o to in , whether the value be posi- 

 tive or negative. But if we assume j- = sin. 0, we must be 



sure that a b, for otherwise ir7 1> or sin. 071, which 

 is impossible. 



(6) The following case is one that frequently occurs 

 x = a sin. A + 6 cos. A. 



Assume = tan. 

 a 



then x = a (sin. A + _ cos. A) 



a 



= a (sin. A + tan. cos. A) 

 ^ n sin. A cos. + sin. cos. A 

 cos. 

 sin. 



(e) Again, if we have 



Assume = tan. 

 a 



cos. 



1 tan. 



1 + tan. " 

 cos. ' sin. ' 

 cos. " + sin. 2 

 = cos. *0 sin. 2 0=cos. 20. 



N.B. It will frequently happen in calculations that 

 we have previously used certain logarithms, and when 

 this is the case the calculation is very materially 

 shortened. Thus, in the above example, suppose we 

 already know log. a and log. 6. Then 



L. tan. 0=log. 6 log. a + 10. 



which immediately enables us to find L. cos. 2 9 ; and 

 therefore x, by only using the tables twice. 



(d) Sin. A=cos. B cos. C, cos. o + sin. B sin. 0. 



u03 - Cc s - tan. 



Assume 

 We obtain 



sin. C. 



sin. A= 



If x 



Assume e = sin. 0. Then 



(g) Such an example as the following frequently occurs 

 in Astronomy : 



If x = m cos. + n cos. (0 + a). Express x in the 

 form A. cos. (0 + B). 



x = m cos. + n cos. a cos. n sin. a sin. 

 = (m + n cos. a) cos. n sin. a sin. 



sin. sin. 0\ 



0~~ ~J 



nCOS - a COS. (0 + 0). 



. /COS. COS. 



(in + n cos. o) I 



COS. 



where tan. 



m + n cos. a 



It will be observed that the expression for tan. 0, is 

 not expressed in products and quotients only ; to effect 

 this we must introduce another angle, 0'. Thus : 



Assume tan 0' = cos. o 

 m 



Then m + n cos. o = m (1 + tan. 0^ 

 m (cos. 0' + sin. 0^ 



cos. 0' 

 m (sin. 0' cos. 45+ cos. 0' sin. 45) \^2~ 



cos. 0' 

 _ m sin. (0 7 + 45) /2 



COS. 0' 



, n sin. a cos. 0* 

 " J< ~ msin. (0' + 46)V2 



m V2. sin. (0'+ 45) cos. (0+0 



and x = : 77 



n. sin. a cos. cos. 



Then x is in the form required. 



THE EELATION BETWEEN THE SIDES AND ANGLES OF 

 TRIANGLES. 



In the following articles, abc represent the sides of a 

 triangle, and ABC the angles which they subtend. 



a sin. A 

 (36). To show that in every triangle ^ = gi^g 



Let A B C be the triangle, from C draw C N perpendicu- 

 lar to A B. Then whether A be acute or obtuse, 

 Fig. 16. Kg. 17. 



c 



and sin. B * -j^, . 

 C1J 



