Being able to manipulate sums of sinusoids is a useful skillwhen dealing with th

Question

Being able to manipulate sums of sinusoids is a useful skillwhen dealing with the sinusoidal responses of circuits. Because the sum of two sinusoids that have the same frequency isalso a sinusoid at that frequency, there are many expressions thatcorrespond to equivalent sinusoids, but which appear to be quitedifferent. It can be very helpful to recognize some of thesealternative forms. a) Show that the relation    is true by expressing A and Bin terms of K and . b) This expression can be used to go both ways.Express K and in terms of A and B. c)If    show that x(t) can be writtenin the form    Express B and interms of A1, A2, 1, and 2.

Explanation / Answer

ResponseDetails: a) Using the expansion formula cos(A+B)= cosA cosB -sinA sinB Kcos(ot+) = K[cosot cos - sinotsin]                      =Kcos cosot -Ksinsinot                      =Acos(ot )+Bsin(ot ) Where A= Kcos and B= -Ksin b) A2 + B2 = (Kcos)2 +(-Ksin)2     A2 + B2 =K2 (cos2 + sin2 ) But (cos2 + sin2 ) =1    A2 + B2 =K2 K = A2 + B2 and B/A = (-Ksin) /(Kcos) = -tan therefore = -tan-1 (B/A) c) x(t) = A1cos(ot+1) +A2cos(ot+2) x(t) = A1 [cosotcos1- sinotsin1]+A2  [cosotcos2- sinotsin2] use the same principle K = A2 + B2 and B/A = (-Ksin) /(Kcos) = -tan therefore = -tan-1 (B/A) c) x(t) = A1cos(ot+1) +A2cos(ot+2) x(t) = A1 [cosotcos1- sinotsin1]+A2  [cosotcos2- sinotsin2] use the same principle and B/A = (-Ksin) /(Kcos) = -tan therefore = -tan-1 (B/A) c) x(t) = A1cos(ot+1) +A2cos(ot+2) x(t) = A1 [cosotcos1- sinotsin1]+A2  [cosotcos2- sinotsin2] use the same principle ResponseDetails: a) Using the expansion formula cos(A+B)= cosA cosB -sinA sinB Kcos(ot+) = K[cosot cos - sinotsin]                      =Kcos cosot -Ksinsinot                      =Acos(ot )+Bsin(ot ) Where A= Kcos and B= -Ksin b) A2 + B2 = (Kcos)2 +(-Ksin)2     A2 + B2 =K2 (cos2 + sin2 ) But (cos2 + sin2 ) =1    A2 + B2 =K2 K = A2 + B2 and B/A = (-Ksin) /(Kcos) = -tan therefore = -tan-1 (B/A) c) x(t) = A1cos(ot+1) +A2cos(ot+2) x(t) = A1 [cosotcos1- sinotsin1]+A2  [cosotcos2- sinotsin2] use the same principle

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