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PLANE TRIGONOMETRY. 



I. Equations. 



IN the solution of Trigonometrical Equations, it must be 

 remembered that when an equation has been reduced to the 

 forms (1) sin a; = sin a, (2) cos x = cos a, (3) tan x = tan a, the so- 

 lutions are (1) x = mr + ( 1)" a, (2) x = 2nir a, (3) x = mr + a, 

 n denoting an integer positive or negative. 



The formulae most useful in Trigonometrical reductions are 



2 sin A cos B = sin (A + B) + sin (A - 

 2 cos A cosB = cos (A + B) + cos (A 

 2 sin A sin B ~ cos (A B) cos (A 

 and 



-*),) 

 -*), 



+ );> 



A+B A-B 



sin A -f sin Jj = 2, sin cos = 



cos A + cos B = 2 cos x cos ; , 



B-A A+B 



cos A - cos B = 2 sm - - sm - ; 

 2 J 



which enables us to transform products of Trigonometrical func- 

 tions, (sines or cosines) into sums of such functions and con- 

 versely. Thus, to transform 



sin 2 {(P-y)} + sin 2 (y - a) + sin 2 (a -ft) 

 sin 2(y-a) + sin 2 (a-0) = 2 sin (y~P) cos (/? + y- 2a), 



sin 2 (P-y) = 2 sin (0 -y) cos (p-y); 

 .'. sin 2 (/3 - y) + sin 2 (y - a) + sin 2 (a - /?) 



= 2sin(/3-y) {cos(/3-y)-cos(/3+y-2a)} 

 = 2 sin ()3-y) {2 sin(^-a) sin (y-a)} 

 = - 4 sin (P - y) sin (y - a) sin (a - /?). 



