Question

If $m$ and $n$ are the smallest positive integers satisfying the relation $\left(2 C \text { is } \frac{\pi}{6}\right)^{m}=\left(4 C \text { is } \frac{\pi}{4}\right)^{n}$, then $(m+n)$ has the value equal to

Answer 1

$2^{m-2 n} \cdot\left[\cos \frac{m \pi}{6}+i \sin \frac{m \pi}{6}\right]=\left[\cos \frac{n \pi}{4}+i \sin \frac{m \pi}{4}\right]$
For equality $m=2 n$
$\frac{n \pi}{4}=\frac{m \pi}{6}+2 k \pi\,\, k \in I$
Put $m=2 n$
$\frac{n \pi}{4}=\frac{n \pi}{3}+2 k \pi ; -\left(\frac{n \pi}{12}\right)=2 k \pi $
(ignore (-)ve sign)
$n=24 k ; m=48 k ; $
for $m, n$ to be smallest $m+n=72$