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Question

Find the real values of θ for which the complex number 1+i cos θ12i cos θ is purely real.


Solution

Let z=1+i cos θ12i cos θ=1+icos θ12i cosθ×1+2i cos θ1+2i cos θ=1+2i cos θ+i cos θ(1+2i cos θ)12+(2 cos θ)2=1+2i cos θ+i cos θ2cos2θ1+4 cos2θ=12cos2θ+3i cos θ1+4cos2θ=12cos2θ1+4 cos2θ+3 cos θ1+4cos2θ

We know that z is purely real if and only if Imz = 0

 3 cos θ1+4 cos2θ=0

           ( z is given to be purely real)

 3 cos θ=0 cos θ=0 cos θ=cos π2

The general solution is given by

θ =2nπ±π2, nϵZ


Mathematics
RD Sharma
Standard XI

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