92 ELEMENTS OF ELECTRICAL ENGINEERING. 



As a second example divide e^ 6 = cos 6 -\-j sin 6 by ir * cos ^ -f / sm ^ ; 

 member by member, and we have : 



or, multiplying numerator and denominator of the right hand member of equa- 

 tion (iv) by cos ty J sin T/>, we have : 



j(0-t) _ cos o cos ^ _j_ sin sin ^ _j_y(sin Ocostjj cos sin i/;) ( v) 



but, according to equation (i) we have : 



so that, comparing equations (v) and (vi), we have 



cos (6 T/;) = cos 6 cos i/> -\- sin 6 sin i/; 

 and sin (6 i/>) = sin 6 cos i/> cos sin ^ 



As a third example the reader may find the expressions for sine and cosine of 

 3$ by cubing both members of equation (i), or he may find the expressions for 

 sine and cosine of (6 -j- i/> -{- ^) by multiplying together the expressions for J , 

 e^ and e^. 



(b] Regarding the formulas of Art. 42. Let z be the numerical value of the 

 impedance of an alternating current circuit, that is, let z=v ~JR* -\- J 2 , and let 

 6 be the angle whose tangent is X\R, then 



(vii) 



Furthermore, let / be the numerical value of the current and let a be the angle 

 between the current and the .r-axis of reference, then 



The expression for the electromotive force is then found by multiplying (vii) 

 by (viii), member by member, and we have : 



(ix) 



That is, the numerical value of is zl, and E is 6 ahead of the current 

 in phase. On the other hand, if we divide equation (ix) by equation (vii) 

 member by member, we get the correct expression for the current. 



(c) Regarding the products of current by current, and current by voltage. 

 If the impedance of a circuit (ae j ' d = J?-\-jX) be looked upon as a vector it 

 must be considered as a stationary vector making the angle with a fixed axis 

 of reference ; whereas the angle between the electromotive force vector and a 

 fixed axis is u(, and the angle between the current vector and a fixed axis is 

 (w/ 6). Therefore, we may write, with reference to a fixed axis : 



Impedance = ze^ 6 

 Voltage = $e* wt 

 and Current = e ^ w <-^ 



