Question

Express the following in a + ib form:

(i) ${\left(\frac{\cos \theta +i\sin \theta }{\sin \theta +i\cos \theta }\right)}^4=\cos n\theta +i\sin n\theta$. Find the value of $n$.

Answer 1

${\left(\frac{\cos \theta +i\sin \theta }{\sin \theta +i\cos \theta }\right)}^4$

$=\frac{(\cos \theta +i\sin \theta {)}^4}{i^4(\cos \theta -i\sin \theta {)}^4}$

$=\frac{(\cos \theta +i\sin \theta {)}^4}{(\cos \theta +i\sin \theta {)}^{-4}}$

$=(\cos \theta +i\sin \theta {)}^8$
$=\cos 8\theta +i\sin 8\theta$
Therefore, $n=8$