Question: Realise the denominator 3+2isin⁡θ1−2isin⁡θ and hence find θ if the expression is purely imaginary.I've realised the denominator 3+8isin⁡θ−4sin2θ1+4sin2θ but don't know how to utilise the knowledge that the expression is purely imaginary.

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Question: Realise the denominator 3+2isin⁡θ1−2isin⁡θ and hence find θ if the expression is purely imaginary.I&#039;ve realised the denominator 3+8isin⁡θ−4sin2θ1+4sin2θ but don&#039;t know how to utilise the knowledge that the expression is purely imaginary.

crazyrocketrz5z · Accepted Answer

The second phrase you used can be expressed as3+8isin(θ)-4sin(θ)21+4sin(θ)2=3-4sin(θ)21+4sin(θ)2⏟real part+i8sin(θ)1+4sin(θ)2⏟imaginary partThe genuine part of a phrase must be 0 if it is to be entirely fictitious.This only happens when the real part&#039;s numerator is 0, or when3−4sin⁡(θ)2=0Via some manipulation, this becomessin(θ)=±32⇒θ∈{±π3+2kπ&amp;nbsp;|&amp;nbsp;k∈ℤ}∪{±2π3+2kπ&amp;nbsp;|&amp;nbsp;k∈ℤ}where k is an arbitrary integer

Question: Realise the denominator \(\displaystyle{\frac{{{3}+{2}{i}{\sin{\theta}}}}{{{1}-{2}{i}{\sin{\theta}}}}}\) and hence

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