THE EQUATION a sin 6 + 6 cos 6- e 33 



( 



When the fraction .- is positive, the angle 6 4- a has two 



Va 2 + b- 



values, one between and 90 and the other between 270 and 

 360. When the fraction b negative, the angle + a has two 

 values, one between 90 and 180 and the other between 180 

 and 270. 



Solve the equation 9 cos 14 sin 6 = 15. 



9 14 15 



Then cos . sin . = - 



V277 A/277 \/277 



cos . cos a sin . sin a = - 0-9014 

 cos (6 + a) = - 0-9014 



tan a = = 1-5555 



7 



+ a = 154 20' or 205 40' 

 a = 57 16' 

 = 97 4' or 148 24' 







It should be noticed that the nature of the fraction . . . 



Va 2 + b 2 



decides upon the possibilities of the equation, for since the value 

 of the fraction represents the sine or cosine of an angle, when 

 c > Va 2 + b 2 , the fraction is greater than 1 and there is no solution 

 to the equation, since the sine or cosine can never be greater 

 than 1 or less than 1. 



If c< Va 2 + b 2 the fraction is less than 1 and the equation has 

 two roots. 



If c = Va 2 + b 2 the fraction is equal to 1 and the equation has 

 one root, for 



(1) Sin (0 + p) = 1 and = 90 - p 



(2) Sin (0 +.p) = - 1 and = 270 - (i 



(3) Cos (0 + a) = 1 and = 360 - a 



(4) Cos (0 + a) = - 1 and = 180 - a 



When c = the equation can be solved in a much simpler way, 

 for if a sin b cos = 



then a sin = T b cos 



and tan qp - giving the values of at once. 



B 



20. The Multiple Angles. The relations for sin (A + B) and 

 cos (A + B) can be used for expressing the trigonometrical ratios 

 of multiple angles of a given angle in terms of the trigonometrical 

 ratios of that angle. 



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



c 



