Question

$\frac{{(\cos \theta +i\sin \theta )}^4}{{(\sin \theta +i\cos \theta )}^5}$ is equal to ......

• A. $\cos 9\theta -i\sin 9\theta$
• B. $\cos 9\theta +i\sin 9\theta$
• C. $\sin 9\theta -i\cos 9\theta$
• D. $\sin 9\theta +i\cos 9\theta$

Answer 1

Answer: (C)

$\sin 9\theta -i\cos 9\theta$

We have ${(\sin \theta +i\cos \theta )}^5={(-i^2\sin \theta +i\cos \theta )}^5$

$=i{(\cos \theta -i\sin \theta )}^5=i.e^{-5\theta }$

Formula: $cos\theta +isin\theta =e^{i\theta }$

$\therefore L.H.S.=\frac{e^{4\theta }}{i.e^{-5\theta }}=-i{e}^{9\theta }$

$=-i(\cos 9\theta +i\sin 9\theta )$

$=\sin 9\theta -i\cos 9\theta$

Ans: C