Question

The value $x$ and $y$ where $\frac{(\cos \theta +i\sin \theta {)}^7}{(\sin \theta +i\cos \theta {)}^4}=x+iy$ is

• A. $x=\sin 11\theta ,y=\cos 11\theta$
• B. $x=\sin 3\theta ,y=\cos 3\theta$
• C. $x=\cos 3\theta ,y=\sin 3\theta$
• D. $x=\cos 11\theta ,y=\sin 11\theta$

Answer 1

The correct option is D $x=\cos 11\theta ,y=\sin 11\theta$

$\frac{(cos\theta +isin\theta {)}^7}{i^4(cos\theta +\frac{sin\theta }{i}{)}^4}=\frac{(cos\theta +isin\theta {)}^7}{(cos\theta -isin\theta {)}^4}=\frac{(cos\theta +isin\theta {)}^7}{\left[cos\right(-\theta )+isin(-\theta ){]}^4}$

$=\frac{(cos\theta +isin\theta {)}^7}{[cos\theta +isin\theta {]}^{-4}}=(\cos 11\theta +i\sin 11\theta )$

Comparing with $x+iy$, we get
$x=\cos 11\theta$
$y=\sin 11\theta$