52 PRACTICAL MATHEMATICS 



1 a sin pt + b cos pt 



Then j-. operating on sm pt gives -3 ^ C 



abi a 2 + b 2 



Classifying these results we get : 



(1) a + bi operating on sin pt gives a sin pt + b cos pt 



or V a 2 + b 2 - sin (pt + a) 



(2) a bi operating on sin pt gives a sin pt b cos pf 



or Va 2 + fe 2 sin (pt a) 



(3) - -r. operating on sin pt gives ^ ^ { sm pt b cos p} 





(4) -- j-. operating on sin pf gives -^ p {a sin pt + b cos 



In each case tan a = - 

 a 



Example. The voltage applied at the sending end of a long 

 telephone line being v sin qt, the current entering the line is 



Vs + ikq . 

 - a* sm qt 

 r + ilq 



where per unit length of cable, r is resistance, I is inductance, 

 s is leakance, and k is permittance or capacity. 



If r = 6 ohms, I = 3 x 10~ 3 henries, k = 5 x 10~ 9 farads, 

 s = 3 x 10~ 6 mho, and is q = 6 x 10 3 , find the current. 



^.-w.^ J^Q 



s + ikq = vs 2 + k z q 2 (cos a + * sin a} where tan a = ~ 



S 



_ Id 



r + ilq = Vr 2 + I 2 q 2 {cos (3 + i sin (3} where tan [3 = 



. _ i 



Then J* +ikq - v f^+^VX - (cos a + sin a 

 \ r + z7 ? " " Uo lr 2 + Z 2 g 2 ) \cos [3 + i sin p 



1 

 = AP O {COS (a - P) + t sin (a - (3) }* where A = 



a- p 



2 Sln 



