32 



Similarly the least value is Va 2 + b 2 , and this occurs when 

 has the values B, B, etc., at intervals of 2?r. 



Also y vanishes when sin (0 + B) = 0. 



That is, when + B has the values 0, TT, 2?r, etc., or when 

 has the values B, TT B, 2?r 8, etc., at intervals of ?r. 



This method of treatment can be applied to the other three cases. 



19. The equation a sin + b cos = c. 



Dividing throughout by Va 2 + b 2 , the equation becomes 



(1) gives the values of the angle + (3, (2) gives the value of 

 the angle B, and the values of can be found by subtraction. 



f> 



When the fraction . = is positive, the angle + B has two 



Va 2 + b 2 



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

 180. When the fraction is negative, the angle + B has two 

 values, one between 180 and 270 and the other between 270 

 and 360. 



Solve the equation 8 sin + 11 cos = 12. 



TU 8 ft n 12 



Then sm . + cos . = 7= 



Vl85 Vl85 A/185 



sin . cos B + cos . sin B = - 0-8822 

 sin (0 + B) = - 0-8822 



tan B = ^ = 1-3750 



8 



+ B = 241 55' or 298 5' 

 B = 53 59' 

 = 187 56' or 244 6' 



The equation b cos a sin = c. 



Dividing throughout by Va 2 + b 2 , the equation becomes 



b . a c 



cos . 



(1) gives the values of the angle + a. 



(2) gives the value of the angle a. 



