Question #6d615

May 28, 2016

(2) 4

Explanation:

The numbers $0$, $1$ and $1 + \sin \left(\frac{\pi}{8}\right) + i \cos \left(\frac{\pi}{8}\right)$ form an isoceles triangle with largest angle $\frac{5 \pi}{8}$ and smaller angles each $\frac{3 \pi}{16}$.

The Complex conjugate $1 + \sin \left(\frac{\pi}{8}\right) - i \cos \left(\frac{\pi}{8}\right)$ forms a similar isoceles triangle with $0$ and $1$.

So the quotient forms an angle $\frac{3 \pi}{8}$ with the Real axis.

Note that, though we are not really interested in the modulus, it will be $1$ since we are dividing a number by its Complex conjugate.

So the quotient is actually $\cos \left(\frac{3 \pi}{8}\right) + i \sin \left(\frac{3 \pi}{8}\right)$

The smallest integer multiplier that makes this angle into an odd multiple of $\frac{\pi}{2}$ is $4$, giving $\frac{3 \pi}{2}$, hence the answer is number (2) $4$

The actual value will be $- i$.