In this question, a, b and θ are all real numbers.(a) Express cos θ in terms of e^iθ and e^-iθ. By taking?

. In this question, a, b and θ are all real numbers.


(a) Express cos θ in terms of e^iθ and e^-iθ. By taking


θ =a – b/2


;


deduce that


e^ia + e^ib = 2e^i(a+b)/2 cos (a – b/2)


(b) Hence show that


Ae^i(ωt+θ1) + Ae^i(ωt+θ2) = 2Ae^i(ωt+Ф1) cos Ф2;


where A, ω, θ1 and θ2 are all real and


Ф1 =θ1 + θ2/2


and Ф2 =


θ1 - θ2/2


:


(c) At time t, I1(t) = Asin(ωt + θ1) and I2(t) = Asin(ωt + θ2) are alternating currents


connected in parallel in a circuit. The load current at time t is given by the relation


IL(t) = I1(t) + I2(t):


Use equation (1) to show that


IL(t) = 2Acos Ф2 sin(ωt + Ф1):

In this question, a, b and θ are all real numbers.(a) Express cos θ in terms of e^iθ and e^-iθ. By taking?
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