Question

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

Answer 1

Answer: 72

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