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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank De Moivre's theorem and Roots of unity

  • question_answer
    If  \[{{\left( \frac{1+\cos \theta +i\sin \theta }{i+\sin \theta +i\cos \theta } \right)}^{4}}=\cos n\theta +i\sin n\theta \], then \[n\] is equal to [EAMCET 1986]

    A) 1

    B) 2

    C) 3

    D) 4

    Correct Answer: D

    Solution :

    \[{{D}^{r}}=i(1+\cos \theta )+\sin \theta =2i{{\cos }^{2}}\frac{\theta }{2}+2\sin \frac{\theta }{2}\cos \frac{\theta }{2}\] \ L.H.S \[={{\left[ \frac{\cos (\theta /2)+i\sin (\theta /2)}{i\cos (\theta /2)+\sin (\theta /2)} \right]}^{4}}\]                      = \[\frac{1}{{{i}^{4}}}{{(\cos \theta +i\sin \theta )}^{4}}=\cos 4\theta +i\sin 4\theta \].


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