30 PRACTICAL MATHEMATICS 



sin (A + B) 



Tan (A + B) - 



cos (A + B) 



sin A cos B + cos A sin B 



cos A cos B sin A sin B 



tan A + tan B 



= T =r on dividing numerator and de- 



1 tan A tan B 



nominator by cos A cos B. 



. sin (A - B) 



Also tan (A - B) - 77 



cos (A B) 



sin A cos B cos A sin B 

 cos A cos B + sin A sin B 



tan A tan B 



= 7 T ^ on dividing numerator and 



1 + tan A tan B 



denominator by cos A cos B. 



These relations for the trigonometrical ratios of the angles 

 (A + B) and (A B) are of utmost importance, and they should 

 be treated as fundamental relations, since so much of the higher 

 work in Trigonometry depends upon them. 



17. The expression a sin + b cos 0. By comparing the ex- 

 pression a sin + b cos with the relations 



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

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



we are enabled to put it as either a sine function or a cosine 

 function, according as to whether the algebraic signs of a and b 

 are alike or unlike. 



(1) When the signs are alike 



a sin + b cos = Va 2 + 2 {sin . + cos .] 



Va 2 + b 2 Va 2 + b 2 f 



This converts the quantities a and b into fractions which are 

 trigonometrical ratios of the angle (3, the base angle of a right- 

 angled triangle whose perpendicular is b and whose base is a. 

 (Fig. 12.) 



The expression thus becomes 



A/a 2 + 6 2 {sin cos (3 + cos sin (3) 



and finally, Va 2 + b 2 sin (0 + B) where tan B = - 



a 



If both the signs are negative, then 

 a sin 6 cos = (a sin 0+6 cos 0) 



= - Va 2 + b 2 sin (0 + B) where tan B = - 



a 



(2) When the signs are unlike 



b cos - a sin = Va 2 +6 2 (cos . b - sin . a \ 



Va 2 + b 2 Va 2 + b 2 ! 



