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Question
Find real values of θ for which `((4 + 3"i" sintheta)/(1 - 2"i" sin theta))` is purely real.
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Solution
Let z = `(4 + 3"i" sintheta)/(1 - 2"i" sintheta)`
= `(4 + 3"i" sintheta)/(1 - 2"i" sintheta) xx (1 + 2"i" sintheta)/(1 + 2"i" sintheta)`
= `(4 + 8"i" sintheta + 3"i" sintheta + 6"i"^2 sin^2theta)/(1 - 4"i"^2 sin^2theta)`
= `(4 + (11 sintheta)"i" - 6 sin^2theta)/(1 + 4 sin^2theta)` ...[∵ i2 = – 1]
= `((4 - 6 sin^2theta) + (11 sintheta)"i")/(1 + 4 sin^2theta)`
∴ z = `((4 - 6 sin^2theta)/(1 + 4 sin^2theta)) + ((11 sintheta)/(1 + 4 sin^2theta))"i"`
Since z is purely real, Im(z) = 0
∴ `(11 sintheta)/(1 + 4 sin^2theta)` = 0
∴ sin θ = 0 = sin nπ, where n ∈ Z
∴ θ = nπ, where n ∈ Z.
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