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  • I need help with - Complex numbers and quadratic equations - JEE Main-4

\frac{\left (\cos \Theta +i\sin \Theta \right )^{4} }{\left (\sin \Theta +i\cos \Theta \right )^{5}}   equals

  • Option 1)

    \sin 9\Theta -i\cos 9\Theta

  • Option 2)

    \sin 9\Theta +i\cos 9\Theta

  • Option 3)

    \sin \Theta +i\cos \Theta

  • Option 4)

    \cos \Theta+i\sin \Theta

 

Answers (1)

\frac{\left ( \cos \Theta +i\sin \Theta \right )^{4}}{\left ( \sin \Theta +i\cos \Theta \right )^{5}}=\frac{\cos 4\Theta +i\sin 4\Theta }{i^{5}\left ( \cos \Theta -isin\Theta \right )^{5}}=\frac{-i\left ( e^{i4\Theta } \right )}{e^{-i5\Theta }}=-ie^{i9\Theta }=-1\left ( \cos 9\Theta +i\sin 9\Theta \right )=\sin 9\Theta -i\cos 9\Theta

 

De Moivre's Theorem -

\left ( \cos \Theta +i\sin \Theta \right )^{n}=\cos n\Theta +i\sin n\Theta \; \forall \; n\, \epsilon I

If n\, \epsilon \, Q , n\, \neq \, I, n=\frac{p}{q},  HCF\left ( p,q \right )=1 then \left ( \cos \Theta +i\sin \Theta \right )^{n} has \left | q \right | number of results, out of which one result will be \cos n\Theta +i\sin n \Theta.

-

 

 


Option 1)

\sin 9\Theta -i\cos 9\Theta

This is correct

Option 2)

\sin 9\Theta +i\cos 9\Theta

This is incorrect

Option 3)

\sin \Theta +i\cos \Theta

This is incorrect

Option 4)

\cos \Theta+i\sin \Theta

This is incorrect

Posted by

subam

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